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Topology and its Applications 155 (2008) 749–751

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Preface

This special issue of Topology and Its Applications is entirely dedicated to the theory of continuous selections ofmultivalued mappings. Since the pioneering work of Ernest Michael from 1956 can rightfully be considered as theyear of birth of this theory, the reader is reading a special issue of the journal dedicated to the 50th anniversary of thetheory of continuous selections. At the same time all papers of this issue are dedicated to the 80th anniversary of thefounder of this theory, Ernest Michael.

The material is divided in two parts. The second part, as is customary, consists of invited original research paperswhich were solicited for this special issue from experts in this area. All these results are in one way the other connectedwith multivalued mappings and their selections. The first part, on the other hand, contains papers which are personallydedicated to Ernest Michael.

It is our pleasure to acknowledge the support of the Topology and its Applications, in particular, Editor-in-ChiefJan van Mill, without whose support and encouragement this issue would have never been completed.

Guest EditorsD. Repovš

P.V. Semenov

0166-8641/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.topol.2007.10.004

750 Topology and its Applications 155 (2008) 749–751

Ernest A. MichaelComplete Bibliography

1. Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951) 152–182.2. Locally multiplicatively-convex topological algebras, Memoirs Amer. Math. Soc. 11 (1952) 79 p.3. Transformations from a linear space with weak topology, Proc. Amer. Math. Soc. 3 (1952) 671–676.4. Some extension theorems for continuous functions, Pacific J. Math. 3 (1953) 789–806.5. A note on paracompact spaces, Proc. Amer. Math. Soc. 4 (1953) 831–838.6. Local properties of topological spaces, Duke Math. J. 21 (1954) 163–171.7. Selection theorems for continuous functions, Proc. Intern. Congr. Math. 2 (1954) 241–242.8. Point-finite and locally finite coverings, Canadian J. Math. 7 (1955) 275–279.9. On local and uniformly local topological properties, Proc. Amer. Math. Soc. 7 (1956) 304–307 (with J. Dugundji).

10. Selected selection theorems, Amer. Math. Monthly 63 (1956) 233–238.11. Continuous selections I, Ann. of Math. 63 (1956) 361–382.12. Continuous selections II, Ann. of Math. 64 (1956) 562–580.13. Continuous selections III, Ann. of Math. 65 (1957) 375–390.14. Another note on paracompact spaces, Proc. Amer. Math. Soc. 8 (1957) 822–828.15. A theorem on semi-continuous set-valued functions, Duke Math. J. 26 (1959) 647–651.16. Dense families of continuous selections, Fund. Math. 47 (1959) 173–178.17. Paraconvex sets, Math. Scand. 7 (1959) 372–376.18. Convex structures and continuous selections, Canadian J. Math. 11 (1959) 556–575.19. Yet another note on paracompact spaces, Proc. Amer. Math. Soc. 10 (1959) 309–314.20. A class of partially ordered sets, Amer. Math. Monthly 67 (1960) 448–449.21. On a theorem of Rudin and Klee, Proc. Amer. Math. Soc. 12 (1961) 921.22. A note on intersections, Proc. Amer. Math. Soc. 13 (1962) 281–283.23. Collared sets, in: General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961), Aca-

demic Press, New York; Publ. House Czech. Acad. Sci., Prague, pp. 270–271.24. The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1963) 375–376.25. Continuous selections in Banach spaces, Studia Math. Ser. Spec. Z.I. (1963) 75–76.26. Completing a spread (in the sense of R.H. Fox) without local connectedness, Indagationes Math. 25 (1963) 629–633.27. Cuts, Acta Math. 111 (1964) 1–36.28. A linear mapping between function spaces, Proc. Amer. Math. Soc. 15 (1964) 407–409.29. Three mapping theorems, Proc. Amer. Math. Soc. 15 (1964) 410–415.30. A short proof of the Arens–Eells embedding theorem, Proc. Amer. Math. Soc. 15 (1964) 415–416.31. Metrizability of certain countable unions, Illinois J. Math. 8 (1964) 351–360 (with H.H. Corson).32. A note on closed maps and compact sets, Israel J. Math. 2 (1964) 173–176.33. On a map from a function space to a hyperspace, Math. Ann. 162 (1965) 87–88.34. ℵ0-Spaces, J. Math. and Mech. 15 (1966) 983–1002.35. A selection theorem, Proc. Amer. Math. Soc. 17 (1966) 1404–1406.36. Separable Banach spaces which admit 1∞

n approximations, Israel J. Math. 4 (1966) 189–198 (with A. Pelczynski).37. Peaked partition subspaces of C(X), Illinois J. Math. 11 (1967) 555–562 (with A. Pelczynski).38. A linear extension theorem, Illinois J. Math. 11 (1967) 563–579 (with A. Pelczynski).39. A note on k-spaces and kR -spaces, in: Proc. Topology Conf. Arizona State Univ., 1968, pp. 247–249.40. Topological well-ordering and continuous selections, Invent. Math. 6 (1968) 150–158 (with R. Engelking and R. Heath).41. Gδ -sections and compact-covering maps, Duke Math. J. 36 (1969) 125–127.42. Local compactness and cartesian products of quotient maps and k-spaces, Ann. l’Inst. Fourier 18 (1968) 281–286.43. Bi-quotient maps and cartesian products of quotient maps, Ann. l’Inst. Fourier 18 (1968) 287–302.44. Quotients of the space of irrationals, Pacific J. Math. 28 (1969) 629–633 (with A.H. Stone).45. Paracompactness and the Lindelöf property for Xn and Xω , in: Proc. Washington State Univ. Conf. on General Topol., Pi

Mu Epsilon, Dept. of Math., Washington State Univ., Pullman, Wash., 1970, pp. 11–12.46. On Nagami’s �-spaces and some related matters, in: Proc. Washington State Univ. Conf. on General Topol., Pi Mu Epsilon,

Dept. of Math., Washington State Univ., Pullman, Wash., 1970, pp. 13–19.47. A theorem on perfect maps, Proc. Amer. Math. Soc. 28 (1971) 633–634.48. A property of the Sorgenfrey line, Comp. Math. 23 (1971) 185–188 (with R.W. Heath).49. Paracompactness and the Lindelöf property in finite and countable cartesian products, Comp. Math. 23 (1971) 199–214.50. On representing spaces as images of metrizable and related spaces, Gen. Topology Appl. 1 (1971) 329–343.51. Spaces with point-countable bases, in: Proc. Univ. Houston Point Set Topology Conf., Univ. Houston, Houston, Tex., 1971,

pp. 37–41.52. A quintuple quotient quest, Gen. Topology Appl. 2 (1972) 91–138.

Topology and its Applications 155 (2008) 749–751 751

53. A new proof of a theorem of V.V. Filippov, Israel J. Math. 11 (1972) 394–397 (with D. Burke).54. On two theorems of V.V. Filippov, in: General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Third

Prague Topol. Sympos., 1971), Academia, Prague, 1972, pp. 307–308.55. Compact-covering images of metric spaces, Proc. Amer. Math. Soc. 37 (1973) 260–266 (with K. Nagami).56. On k-spaces, kR -spaces and k(X), Pacific J. Math. 47 (1973) 487–498.57. Countably bi-quotient maps and A-spaces, in: Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg,

VA, 1973), Lecture Notes Math., vol. 375, Springer, Berlin, 1974, pp. 183–189.58. Some classes of quotient maps, in: Topological Structures (Proc. Sympos. in Honour of Johannes de Groot (1914–1972),

Amsterdam, 1973), Math. Centre Tracts, vol. 52, Math. Centrum, Amsterdam, 1974, pp. 55–58.59. Quotients of countable complete metric spaces, Proc. Amer. Math. Soc. 57 (1976) 371–372.60. On certain point-countable covers, Pacific J. Math. 64 (1976) 79–92 (with D. Burke).61. A-spaces and countably bi-quotient maps, Dissert. Math. 133 (1976) 5–48 (with R.C. Olson and F. Siviec).62. Barely continuous functions, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 24 (1976) 889–892 (with I. Namioka).63. ℵ′

0-spaces and a function space theorem of R. Pol, Indiana Univ. Math. J. 26 (1977) 299–306.64. σ -locally finite maps, Proc. Amer. Math. Soc. 65 (1977) 159–164.65. Complete spaces and tri-quotient maps, Illinois J. Math. 21 (1977) 716–733.66. A note on Eberlein compacts, Pacific J. Math. 72 (1977) 487–495 (with M.E. Rudin).67. Another note on Eberlein compacts, Pacific J. Math. 72 (1977) 497–499 (with M.E. Rudin).68. Uniform AR’s and ANR’s, Comp. Math. 39 (1979) 129–139.69. Some results on continuous selections, in: Topological Structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978),

Math. Centre Tracts, vol. 115, Math. Centrum, Amsterdam, 1979, pp. 161–163.70. A unified theorem on continuous selections, Pacific J. Math. 87 (1980) 187–188 (with C. Pixley).71. Continuous selections and finite-dimensional sets, Pacific J. Math. 87 (1980) 189–197.72. Continuous selections and countable sets, Fund. Math. 111 (1981) 1–10.73. Inductively perfect maps and tri-quotient maps, Proc. Amer. Math. Soc. 82 (1981) 115–119.74. On maps related to σ -locally finite and σ -discrete collections of sets, Pacific J. Math. 98 (1982) 139–152.75. A result on shrinkable open covers, Topology Proc. 8 (1983) 37–43 (with G. Gruenhage).76. Spaces determined by point-countable covers, Pacific J. Math. 113 (1984) 303–332.77. A parametrization theorem, Topology Appl. 21 (1985) 87–94 (with G. Mägerl and R.D. Mauldin).78. A note on completely metrizable spaces, Proc. Amer. Math. Soc. 96 (1986) 513–522.79. A note on a selection theorem, Proc. Amer. Math. Soc. 99 (1987) 575–576.80. Continuous selections avoiding a set, Topology Appl. 28 (1988) 195–213.81. Continuous selections: a guide for avoiding obstacles, in: General Topology and its Relations to Modern Analysis and Alge-

bra, VI (Prague, 1986), Res. Exp. Math. 16 (1988) 345–349.82. Two questions on continuous selections, Questions Answers Gen. Topol. 6 (1988) 41–42.83. A generalization of a Theorem on Continuous Selections, Proc. Amer. Math. Soc. 105 (1989) 236–243.84. Some problems, in: J. van Mill, G.M. Reed (Eds.), Open Problems in Topology, North-Holland, Amsterdam, 1990, pp. 271–

278.85. Almost complete spaces, hypercomplete spaces and related mapping theorems, Topology Appl. 41 (1991) 113–130.86. Partition-complete spaces are preserved by tri-quotient maps, Topology Appl. 44 (1992) 235–240.87. Some refinements of a selection theorem with 0-dimensional domain, Fund. Math. 140 (1992) 279–287.88. Selection theorems with and without dimensional restriction, in: Recent Developments of General Topology and its Applica-

tions, International Conference in Memory of Felix Hausdorff (1868–1942), Math. Research 67, Berlin, 1992, pp. 218–222.89. Representing spaces as images of 0-dimensional spaces, Topology Appl. 49 (1993) 217–220 (with M.M. Choban).90. A note on bi-quotient and tri-quotient maps, Bull. Polish Acad. Sci. 41 (1993) 285–291.91. A note on global and local selections, Topology Proc. 18 (1993) 189–194.92. Partition-complete space and their preservation by tri-quotient and related maps, Topology Appl. 73 (1996) 121–131.93. Partition-complete paracompact k-space are preserved by closed maps, Topology Appl. 96 (1999) 63–73.94. J -spaces, Topology Appl. 102 (2000) 315–339.95. A survey of J -spaces, in: Proc. Toposym 2001, Topology Atlas, pp. 191–193.96. Closed retracts and perfect retracts, Topology Appl. 121 (2002) 451–468.97. A theorem of Nepomnyashchii on continuous subset-selections, Topology Appl. 142 (2004) 235–244.98. Continuous Selections, Encyclopedia of General Topology, c-8 (2004) 107–109.99. Paracompact spaces, Encyclopedia of General Topology, d-12 (2004) 195–197.

100. A note on convex Gδ -subsets of Banach spaces, Topology Appl. 155 (8) (2008) 858–860 (with I. Namioka).



A mathematical friendship

Isaac Namioka has written a brief article in this Journal on the life of Ernie Michael whom we hope to honor. Itreflects Ernie’s preference for clear, formal exposition and ends as Ernie in 1953 arrives at the University of Wash-ington where he remains, becoming a premier General Topologist as well as Namioka’s friend, colleague, closestmathematical confidant. Thus, for Namioka, “The rest is history”.

I married Walter Rudin in 1953 so this is also a historic year for me. But a year or two earlier, when I was single andErnie was at the Institute for Advanced Studies, Ernie and I first met and discovered the similarities and differencesin our mathematical interests and preferences. We both enjoyed the stimulation of discussing what was going on inGeneral Topology which was becoming our field.

These discussions became the pattern of our friendship. Ernie and I had no joint papers; we never worked on thesame problem at the same time; we never discussed the details of a proof or construction. Ernie was much more awareof what was going on in other countries and other fields, he was better read, more apt to analyze, and infinitely bettereducated. We seldom saw each other; but our friendship was quite important to both of us.

We both wanted both the spaces we worked with and the theorems we tried to prove to be simple to describe andunderstand, and with plenty of meat on them to make the results of some interest to mathematicians outside of GeneralTopology. Ernie was a structure builder: an expert at seeing those details which could be used effectively in buildinga useful theory. He found paracompactness fascinating. I admit I most enjoy tearing down structures: proving thatreasonable sounding conjectures are false: constructing counterexamples. Normality is my thing since it seldom hasthe strength to guarantee your space is not miserable in some other way.

Let us call a space Dowker if it is normal but its product with an ordinary closed unit interval is not normal.Hugh Dowker in England (and Kitti Morita in Japan) had proved that Dowker spaces are the normal non-countablyparacompact spaces. Ernie told me this and that it was unknown if there was a Dowker space. So here was a problemof interest to both of us. Ernie would have preferred for there to be no Dowker space while I was already happilysearching for one! I already knew that it was also unknown if there existed a Souslin line and had some experiencewith its combinatorics and soon after Ernie mentioned the Dowker spaces problem to me, I proved that if there is aSouslin line there is a Dowker space. The paper proving this was published in 1955 when there was a month longtopology conference in Wisconsin at which Ernie ran a seminar on paracompactness that I enjoyed very much.

In the early 1960s Paul Cohen proved that the Continuum Hypothesis is undecidable in ZFC (the usual axioms forSet Theory). His methods almost immediately led to a proof that the same holds for Souslin lines. So it was known thateither the same was also true for Dowker spaces or there existed a “real” Dowker space. Thus this problem continuedto nag me. Finally, on the train on our way to the International Congress in Nice, France in 1970, I completed theconstruction of a “real” Dowker space. I was very excited and the first person I saw as I stepped off the train wasErnie. He too was full of enthusiasm and congratulations and we had a happy celebration. He was doubly glad to seeme for A.V. Arhangel’skii of Moscow had been unable to come to the Congress and had sent Ernie the manuscript forhis lecture on General Topology. Ernie wanted to leave before the lecture was scheduled, but he felt all would go wellif I spoke instead.

The Dowker space problem certainly had consequences for me. If I wanted to continue to work on the kind ofproblems I was most attached to, I had to learn some set theory and take advantage of its insights even if my problemhad an absolute answer (which I preferred).


M.E. Rudin / Topology and its Applications 155 (2008) 752–753 753

Ernie became and remains the world expert on paracompactness. But I think he was always aware of the settheoretic nature of much of classical General Topology and he had deliberately decided he did not want to becomea logician. Thus he mostly concentrated his talents and insights for structure building in the many varied, less settheoretic, areas of General Topology. Selection Theory which had been of special interest to him since the mid 1950swas certainly one. It had virtues he valued highly: the problems could be quite hard but the answers were valuedand understood by mathematicians in many fields and the subject lent itself to being built into a real theory, a usefulstructure and one he enjoyed.

Since the 1960s Ernie and I have both worked all sorts of different problems although the problems each of us chosehad different flavors. Still we both find the full span of General Topology interesting and enjoy talking about it. Anyexcuse will do. Walter and I have visited the Michaels at their home in Seattle whenever we were nearby: for instanceon our way to the International Congress in Vancouver in 1974. Ernie and I spent a happy afternoon wandering aroundhis birthplace, Zürich, when he showed up briefly during the Congress there in 1994. A very special occasion occurredduring the Congress in Kyoto in 1990 on the evening of Ernie’s 65th birthday. Exactly three couples: the Michaels,the Namiokas, and the Rudins got together to celebrate in the Namioka’s hotel room. It was a wonderful party. Erniewas talking of retiring soon and I was saying he could not because I was older. But let me like Namioka warn you: forsome there is “no retirement from Topology”.

M.E. RudinDepartment of Mathematics,

University of Wisconsin,Van Vleck Hall,

480 Lincoln Drive,Madison, WI 53705-1388,

USAE-mail address: [email protected]

Topology and its Applications 155 (2008) 754


Ernie

Professor Ernest Michael (Ernie) was born in 1925 in Zürich, Switzerland. Almost 50 years later, while visiting theE.T.H. in Zürich, Ernie entered the Kantonsspital for some minor surgery. During the registration formality, a hospitalofficial asked: “Where were you born?”

Ernie replied: “Here.”“What do you mean ‘here’?”“Right here in this hospital.”Eventually Ernie came to the United States in 1939, and in 1942, he entered Cornell University as an engineering

student. After his second year, he served in the U.S. Navy for two years.He returned to Cornell in 1946 as a mathematics major, finishing his B.A. degree in 1947. I think the best thing the

Navy did for Ernie was to give him time to rethink his career and change his major from engineering to mathematics.We should all be grateful for that, and I am not saying this just for mathematics!

Subsequently, he earned his M.A. at Harvard in 1948 and his Ph.D. from the University of Chicago in 1951. Hespent the next two years at the Institute for Advanced Study and the University of Chicago on a postdoctoral fellow-ship, and in 1953 he joined the mathematics department of the University of Washington as an Assistant Professor.“The rest is history” as they say. In 1956 he became an Associate Professor, followed by a Professorship in 1960 untilhis retirement in 1993 as Professor Emeritus. However there is no retirement from topology for Ernie.

Ernie said that he was always interested in mathematics even as a child. He recalls fondly the times when he tookwalks with his father, during which his father would challenge him with various mathematical problems. However, hewas exposed to topology relatively late: it occurred during the first quarter at the University of Chicago in a courseon general topology taught by Edwin Spanier. Ernie was so interested in the subject that he wanted to do his thesisresearch in this area. However, upon being told that it was not possible to receive a degree at the University of Chicagoin that field, Ernie switched his focus to working in functional analysis with Irving Segal. His doctoral dissertation,titled Locally multiplicatively-convex topological algebras, was published as A.M.S. Memoirs, No. 11, in 1952. AsErnie puts it, this was his last effort in functional analysis. After receiving his Ph.D. in 1951, he has become a full-timegeneral topologist. And this is how we all know him today.

Isaac NamiokaUniversity of Washington,

Department of Mathematics,Box 354350, Seattle,

WA 98195-4350, USAE-mail address: [email protected]




Ernest Michael and theory of continuous selections

Dušan Repovš a,∗, Pavel V. Semenov b

a Institute of Mathematics, Physics and Mechanics, and Faculty of Education, University of Ljubljana, Jadranska 19, P.O. Box 2964,Ljubljana, Slovenia 1001

b Department of Mathematics, Moscow City Pedagogical University, 2-nd Selskokhozyastvennyi pr. 4, 129226 Moscow, Russia

Received 14 June 2006; accepted 15 June 2007

MSC: primary 54C60, 54C65, 41A65; secondary 54C55, 54C20

Keywords: Multivalued mapping; Upper semicontinuous; Lower semicontinuous; Convex-valued; Continuous selection; Approximation; Vietoristopology; Banach space; Fréchet space; Hyperspace; Hausdorff distance

To follow the thoughts of a great man is the most interesting science.A.S. Pushkin

1. Introduction

For a large number of those working in topology, functional analysis, multivalued analysis, approximation theory,convex geometry, mathematical economics, control theory, and several other areas, the year 1956 has always beenstrongly connected with the publication by Ernest Michael of two fundamental papers on continuous selections whichappeared in the Annals of Mathematics [4,5].

With sufficient precision that year marked the beginning of the theory of continuous selections of multivaluedmappings. In the last fifty years the approach to multivalued mappings and their selections, set forth by Michael [4,5],has well established itself in contemporary mathematics. Moreover, it has become an indispensable tool for manymathematicians working in vastly different areas.

Clearly, the principal reason for this is the naturality of the concept of selection. In fact, many mathematicalassertions can be reduced to using the linguistic reversal “∀x ∈ X ∃y ∈ Y . . .”. However, as soon as we speak of thevalidity of assertions of the type

∀x ∈ X ∃y ∈ Y P (x, y)

it is natural to associate to every x a nonempty set of all those y for which P(x, y) is true. In this way we obtain amultivalued map which can be interpreted as a mapping, which associates to every initial data x ∈ X of some problemP a nonempty set of solutions of this problem

F :x �→ {y ∈ Y : P(x, y)

}, F :X → Y.

* Corresponding author.E-mail addresses: [email protected] (D. Repovš), [email protected] (P.V. Semenov).


756 D. Repovš, P.V. Semenov / Topology and its Applications 155 (2008) 755–763

The question of the existence of selections in such a setting turns out to be the question about the unique choice ofthe solution of the problem under given initial conditions. Different types of selections are considered in differentmathematical categories.

One could say that the key importance of Michael’s theory is not so much in providing a comprehensive solutionof diverse selection problems in the category of topological spaces and continuous maps, but rather the immediateinclusion of the obtained results into the general context of development of topology. In a remarkable number ofcases, results of Michael on solvability of the selection problems turned out to be the final answers, i.e. they providedconditions which turned out to be necessary and sufficient.

Initially we were planning to write a survey paper, which would present the development of the theory in the lasthalf of the century and its many applications. However, already our first attempts at such a project showed that thevolume of such a survey would invariably fill an entire book, hence it would be inappropriate for this special issue.

After some deliberations we decided to limit ourselves to a survey of only the papers of Michael on the theory ofselections and their mutual relations. For analogous reasons we do not give any precise references to many develop-ments in the theory of selections—the number of papers in this area is by now around one thousand. A considerablenumber of facts on selections and theorems, which go beyond the present paper, can be found in books and surveys[R1–R15] listed at the end of the paper.

2. Bibliography

Papers in scientific journals usually end with the list of references. In our opinion, is it most reasonable to begin asurvey dedicated to the work of a single person, on one special topic, spanning over 50 years, with a complete list ofhis papers on the subject.

LIST OF ALL PAPERS BY E. MICHAEL ON SELECTIONS

1. Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951) 152–182.2. Selection theorems for continuous functions, Proc. Int. Congr. Math. 2 (1954) 241–242.3. Selected selection theorems, Amer. Math. Monthly 63 (1956) 233–238.4. Continuous selections I, Ann. of Math. (2) 63 (1956) 361–382.5. Continuous selections II, Ann. of Math. (2) 64 (1956) 562–580.6. Continuous selections III, Ann. of Math. (2) 65 (1957) 375–390.7. A theorem on semi-continuous set valued functions, Duke Math. J. 26 (1959) 647–652.8. Dense families of continuous selections, Fund. Math. 47 (1959) 174–178.9. Paraconvex sets, Math. Scand. 7 (1959) 372–376.

10. Convex structures and continuous selections, Canadian J. Math. 11 (1959) 556–575.11. Continuous selections in Banach spaces, Studia Math. Ser. Spec. (1963) 75–76.12. A linear mapping between function spaces, Proc. Amer. Math. Soc. 15 (1964) 407–409.13. Three mapping theorems, Proc. Amer. Math. Soc. 15 (1964) 410–415.14. A short proof of the Arens–Eells embedding theorem, Proc. Amer. Math. Soc. 14 (1964) 415–416.15. A selection theorem, Proc. Amer. Math. Soc. 17 (1966) 1404–1406.16. Topological well-ordering, Invent. Math. 6 (1968) 150–158 (with R. Engelking and R. Heath).17. A unified theorem on continuous selections, Pacific J. Math. 87 (1980) 187–188 (with C. Pixley).18. Continuous selections and finite-dimensional sets, Pacific J. Math. 87 (1980) 189–197.19. Continuous selections and countable sets, Fund. Math. 111 (1981) 1–10.20. A parametrization theorem, Topology Appl. 21 (1985) 87–94 (with G. Mägerl and R.D. Mauldin).21. A note on a selection theorem, Proc. Amer. Math. Soc. 99 (1987) 575–576.22. Continuous selections avoiding a set, Topology Appl. 28 (1988) 195–213.23. A generalization of a theorem on continuous selections, Proc. Amer. Math. Soc. 105 (1989) 236–243.24. Some problems, in: J. van Mill, G.M. Reed (Eds.), Open Problems in Topology, North-Holland, Amsterdam, 1990,

pp. 271–277.25. Some refinements of a selection theorem with 0-dimensional domain, Fund. Math. 140 (1992) 279–287.26. Selection theorems with and without dimensional restriction, in: Recent Developments of General Topology and its Applica-

tions, International Conference in Memory of Felix Hausdorff (1868–1942), Math. Res. 67, Berlin, 1992.27. Representing spaces as images of 0-dimensional spaces, Topology Appl. 49 (1993) 217–220 (with M.M. Choban).28. A note on global and local selections, Topology Proc. 18 (1993) 189–194.

D. Repovš, P.V. Semenov / Topology and its Applications 155 (2008) 755–763 757

29. A theorem of Nepomnyashchii on continuous subset-selections, Topology Appl. 142 (2004) 235–244.30. Continuous selections, Encyclopedia of General Topology, c-8 (2004) 107–109.

We have selected the papers on selections [4,5,7] to serve as the basis of the classification of the entire list. Here isa reasonably precise diagram of relationship among the papers from the list:

Here the usual arrow means direct correlation and the dotted arrow means an implicit one.Papers [2], [11], [26], [28], [30] are not included in this diagram, since they are either short announcements (or

abstracts) on conferences or they are devoted to popularization of the subject.

3. Papers from 1956

A considerable number of fundamental mathematical papers can be divided into two types. In such papers, as a rule,a significant new theory is constructed or an important problem is solved. This division is of course, conditional—onthe one hand, in constructions of new theories one often encounters difficult problems, on the other hand a solution ofa difficult problem often gives rise to a development of a significant new theory.

The papers [4] and [5] are a clear cut example of such a division. In [4] an essentially new mathematical theoryis constructed, in the form of a branched tree, which unifies a large number of sufficiently different theorems. To thecontrary, in [5] the principal result consists of the proof of a single highly nontrivial theorem and all assertions andconstructions in this paper are devoted to the solution of this problem.

Another, linguistic difference between [4] and [5] is connected with the notion of convexity: the formulations ofpractically all theorems of [4] use the term convex, where to the contrary, this word is practically absent from [5].Finally, in [4] Lebesgue dimension is never used, while in [5], there are dimension restrictions on the domains of themultivalued maps everywhere.

One can say, with sufficient accuracy, that in [5] the finite-dimensional, purely topological analogue of such anontopological notions as convexity and local convexity are presented and studied. Without any doubt, the best knownassertion of [4] is Theorem 3.2′′.

Theorem 1. The following properties of a T1-space X are equivalent:

(a) X is paracompact; and(b) If Y is a Banach space, then every lower semicontinuous (LSC) carrier φ :X → Fc(Y ) admits a singlevalued

continuous selection.


Here Fc(Y ) denotes the family of all nonempty closed convex subsets of Y . Observe that in [4] Michael originallyused the term “carrier” instead of “multivalued mapping”. In mathematical practice the implication (a) ⇒ (b) has thewidest application and is in folklore known as the “Convex-Valued Selection Theorem”. The implication (b) ⇒ (a)gives a selection characterization of paracompactness.

The unusual numeration 3.2′′ for the theorem has a very simple explanation. In Chapter 3 of [4] Michael started bya citation of Theorem 3.1 (Urysohn, Dugundji, Hanner) and Theorem 3.2 (Dowker) on the extensions of singlevaluedmappings and then presented the sequences:

Theorem 3.1, Theorem 3.1′, Theorem 3.1′′, Theorem 3.1′′′ and

Theorem 3.2, Theorem 3.2′, Theorem 3.2′′

of their analogs for multivalued mappings. To be more clear, let us unify Theorems 3.1 (a, b, c below) and 3.1′ (a, d,e below) as follows:

Theorem 2. The following properties of a T1-space X are equivalent:

(a) X is normal;(b) The real line R is an extensor for X;(c) Every separable Banach space is extensor for X;(d) Every LSC carrier φ :X → C(R) admits a singlevalued continuous selection; and(e) If Y is a separable Banach space, then every LSC carrier φ :X → C(Y ) admits a singlevalued continuous selec-

tion.

Thus, playing with words, such a series shows that the selection theory in fact, extends the theory of extensors.Here C(Y ) = {Z ∈Fc(Y ): Z is compact or Z = Y }.

As asserted by Michael, Theorem 3.2′′ was his very first selection theorem, the initial goal of which were gener-alizations of a theorem due to R. Bartle and L. Graves on sections of linear continuous surjections between Banachspaces. In particular, Proposition 7.2 of [4] states that such a section can be chosen in an “almost” linear fashion(scalar homogenous) and with the pointwise norm arbitrarily close to the “minimal” of all possible.

Thus the remaining Theorems 3.1′′–3.2′ are selection characterizations of other properties of the domain of aconvex-valued mapping: normality, collectionwise normality, normality and countable paracompactness, and perfectnormality. Many constructions and ideas from [4] later became the basis for subsequent research. For example, Lem-ma 5.2 in [4] was the first result in finding pointwise dense families of selections.

In comparison with [4], the paper [5] originally dealt only with the unique Theorem 1.2, the so-called “Finite-dimensional selection theorem”:

Theorem 3. Let X be a paracompact space, A ⊂ X a closed subset with dimX(X \ A) � n + 1, Y a complete metricspace, F an equi-LCn family of nonempty closed subsets of Y and φ :X → F an LSC map. Then every singlevaluedcontinuous selection of φ|A can be extended to a singlevalued continuous selection of φ|U , for some open subsetU ⊃ A. If additionally every member of F is n-connected (briefly, Cn) then one can take U = X.

Without any doubt, this is one of the most complicated topological theorems, the six-step proof in [5] is clearly amathematical masterpiece. Various efforts were made by several people in the last 50 years to simplify this proof (or“improve” it), including ourselves. However, none of these versions turned out to be shorter or simpler. In our opinion,none of them reached the clarity of exposition in [5].

Two years later, in 1958, Dyer and Hamström applied this theorem to get the sufficient conditions for a regularmap f to be a trivial fibration. Such a condition turned out to be local n-connectedness (LCn) of the homeomorphismsgroup H(M) of the fiber M of f . The problem when H(M) is LCn was one of the central in topology over a periodof almost 20 years and served as one of the key sources for the development of infinite-dimensional topology, as aseparate part of topology.

For the first encounter with the theory of selections, the papers [4] and [5] are too difficult and too voluminous.On the other hand, the short note [3] quickly tells the reader of the most popular method of selection theory—the


method of outside approximation. The note consists of the proof of the Convex-valued and the 0-dimensional selectiontheorems. The last theorem is a particular case of the Finite-dimensional theorem, for n = 0.

Theorem 4. If X is a zero-dimensional (dimX = 0) paracompact space and Y is a complete metric space, then everyLSC mapping φ :X → F(Y ) admits a singlevalued continuous selection.

In spite of its relative simplicity and clarity of its proof, the Zero-dimensional selection theorem has surprisinglymany applications in selection theory and other areas of mathematics.

The last paper of the series [4–6] dealt mainly with restrictions on the displacement of a closed subset A in X. Forexample, as in the Borsuk pairs, when X = Z ×[0;1] and A = (Z × 0)∪ (B ×[0;1]) for an appropriate B ⊂ X. Alsothe lower semicontinuity assumption in [6] was strengthened by continuity in the corresponding Hausdorff metric hρ

in expX. Here we reproduce a typical statement (Theorem 6.1):

Theorem 5. Let X be a paracompact space with dimX � n + 1 and A ⊂ X a weak deformation retract of X. Let(Y,ρ) be a complete metric space, F a uniformly-LCn family of nonempty closed subsets of (Y,ρ) and φ :X → F acontinuous map with respect to hρ . Then every selection of φ|A can be extended to a singlevalued continuous selectionof φ.

Surprisingly, deep constructions and results of [6] have until now had no essential applications.

4. Papers from 1959

We begin by the first paper of the series [7–10]. If one combines arbitrary paracompact domains, as in theConvex-valued selection theorem, and arbitrary complete metric ranges for closed-valued mappings, as in the Zero-dimensional selection theorem, then of course, there is no hope of obtaining a singlevalued continuous selection. Itturned out that under those assumptions a sufficiently fine multivalued selections exist. It was rather an unexpected and“. . .curious result about semi-continuous. . . , [7]” selections. Below, 2Y denotes the family of all nonempty subsets ofa set Y :

Theorem 6. (See [7, Theorem 1.1].) Let X be a paracompact space, Y a metric space, and φ :X → 2Y an LSC mapwith each φ(x) complete. Then there exist ψ :X → 2Y and θ :X → 2Y such that:

(a) ψ(x) ⊂ θ(x) ⊂ φ(x) for all x ∈ X;(b) ψ(x) and θ(x) are compact, for all x ∈ X;(c) ψ is LSC; and(d) θ is USC (upper semicontinuous).

It appears that this was in principle, the very first theorem on multivalued selections. The proof of this Compact-valued selection theorem is based on the so-called method of inner approximations. Roughly speaking, one caninscribe into each value φ(x) a tree with a countable set of levels, with finite sets of vertices on each level, so thateach maximal linearly ordered sequence of vertices will be fundamental.

Thus the sets ψ(x) and θ(x) are constructed as the sets of limits of different kinds of such maximal paths in the tree.Shortly, ψ(x) and θ(x) are limits of certain inverse (countable) spectra in the complete metric space φ(x). Beginningby [7], multivalued selections became by then a fully respected part of general selection theory.

The comprehensive fundamental paper [10] also had an important impact on the development of selection theory.In that paper the axiomatic theory of convexity in metric spaces was presented. As far as we know, this was also oneof the first papers on axiomatic convexities. It served as the starting point for many investigations in this direction.

Also, the method of inner approximations from [7] was changed and applied in [10] to convex-valued maps.Roughly speaking, at each level of a tree above one can consider the barycenter of all vertices at that level, withrespect to a suitable continuous partition of the unity of the domain. In this way it is possible to obtain a pointwiseconvergent sequence of singlevalued (discontinuous!) selections with degree of discontinuity uniformly tending tozero. Therefore the limit gives the desired continuous singlevalued selection.


In our experience, we have encountered several times the situations when the simpler and more direct smoothingmethod of outside approximations did not work, whereas the method of inner approximations successfully solved theproblem at hand. Looking at the data on submission of the papers, one may perhaps infer that [10] was originally thesource for [7].

Whereas Lemmas 5.1 and 5.2 and Theorem 3.1′′′ were proved in [4] for perfectly normal domains and separableBanach range spaces, a version was obtained in [8] for metric domains and any Banach range spaces. The proof wasbased on the replacement of the Gδ-property for closed subsets of a perfectly normal domain by the Stone theoremon the existence of σ -discrete closed basis in any metric space. Note also that Theorem 5.1 [8] on the one hand, usedthe ideas from the proofs in [6], and on the other hand was the basis for the later appearance of such notions as SEPand SNEP (selection extension and selection neighborhood extension properties) in [18].

While [10] estimates the relations and links between convex and metric structures on the set, the paper [9] dealswith the degree of nonconvexity of a closed subset P of a Banach space, endowed with standard convex and metricstructures. Simply put, imagine that we move the endpoints of a segment of length 2r over a set P . In this situation itis very natural to look for the distance between the points of segment and the set P .

So if all such distances are less than or equal to α · r for some constant α ∈ [0;1), then the set P is paraconvexin dimension 1. By passing to triangles, tetrahedra, and n-simplices, one obtains the notion of a paraconvex set. So,as it was proved in [9], the statement of the Convex-valued selection theorem [3,4] holds whenever one replaces theconvexity assumption for the values φ(x) by their α-paraconvexity, for some common α ∈ [0;1), for all x ∈ X.

Moreover, the proof looks as a double sequential “improvement” process of exactness of approximation, on theaccount of applying the Convex-valued selection theorem.

5. Papers from 1964–1979

One of the main purposes of the series [12–15] was to examine the metrizability assumption for the range spacein the Convex-valued selection theorem. In the papers [12–14] improvements of the Arens–Eells embedding theoremwere proved and a selection theorem for mappings from metric domains into completely metrizable subsets of locallyconvex topological vector (LCTV) spaces was established. It was shown in [15] that the statement holds for paracom-pact domains as well. Observe that for LCTV spaces completness is a delicate and in general, “multivalued” notion.Below, a LCTV space is said to be complete if the closed convex hull of any compact subset is also a compact subset.

Theorem 7. (See [15, Theorem 1.2].) Let X be a paracompact space and (M,ρ) a metric subset of a completeLCTV space E. Let φ :X → 2M be an LSC map such that every φ(x) is ρ-complete. Then there exists a continuoussinglevalued f :X → E such that for every x ∈ X, the value f (x) belongs to the closed convex hull of the set φ(x).

Note that one of the key ingredients of the proof is the Compact-valued selection theorem. Next, if φ is convex-valued and closed-valued, then completness of the entire E can be replaced by completness of the closed spansof φ(x), x ∈ X. Such a replacement can also be derived from the Zero-dimensional selection theorem and by thetechnique of pointwise integration (see [R13]).

In the joint paper with Engelking and Heath [16], Michael in some sense returned to his first selection publication[1]. Namely, by using embeddings into closed topologically well-ordered subspaces of the Baire space B(m), theyproved [16, Corollary 2] that for any complete metric, zero-dimensional (with respect to dim or Ind) space (X,ρ)

there exists a singlevalued continuous selector f on the family F(X) of all nonempty closed subsets of X.Here F(X) is endowed with the Hausdorff topology, say τρ , and f :F(X) → X is a mapping with f (A) ∈ A for

every A ∈F(X). The zero-dimensionality is the necessary restriction, because for example, there are no selectors forF(R) (see [16, Proposition 5.1]).

Note that formally, a selector is simply a selection of the multivalued evaluation mapping, which associates to eachA ∈F(X) the same A, but as a subset of X. However, historically the situation was reverse. In [1] Michael proposed adivision of the problem about the existence of a selection g :Y → X for G :Y → 2X into two separate problems: first,to check that G is continuous and second, to prove that there exists a selector on 2X . Hence, the selection problemwas originally reduced to a certain selector problem.


6. Papers from 1980–1990

Pick points x1, x2, . . . , xn in the domain X of a multivalued mapping φ and arbitrary select points f (xi) ∈ φ(xi),using the Axiom of choice. Thus we find a partial selection of φ over the closed subset C = {x1, x2, . . . , xn} ⊂ X.By replacing the values φ(xi) with the singletons {f (xi)} we once again obtain an LSC mapping, say φC . If allassumptions of a selection theorem hold for the new LSC mapping φC , then such a mapping admits a selection, andhence φ also admits a selection.

This simple observation shows that any restrictions for the value of φ over a finite subset C ⊂ X, like closedness,connectivity, convexity, etc. are inessential for the existence of a continuous selection of φ. But what can one say aboutsuch an omission for an infinite C ⊂ X? Clearly, C should be a sufficiently “small”, “dispersed”, etc. subset of X.At the International congress of mathematicians in Vancouver in 1974, Michael announced results for countable C.Based on this, the following result was published in 1981 (see [19, Theorem 1.4]):

Theorem 8. Let X be a paracompact space, Y a Banach space, C ⊂ X a countable subset and φ :X → 2Y an LSCmap with closed and convex values φ(x) for all x /∈ C. Then for every closed subset A ⊂ X, each selection of φ|Aadmits an extension which is a selection of φ (shortly, φ has SEP).

Briefly, over a countable subset of a domain we can simply omit any restriction for the values of LSC mapφ :X → 2Y . A year before, in a joint paper with Pixley [17], Michael proved that the convexity assumption canbe omitted over any subset Z ⊂ X with dimX Z = 0.

Roughly speaking, results of [17–19,23,25] are principally related to several possibilities for relaxing convexity inselection theorems and in particular, the closedness assumptions for values of multivalued mappings. For example, letus mention the following two results:

Theorem 9. (See [19, Theorem 7.1].) Let X be a paracompact space, Y a Banach space, C ⊂ X a countable subset,Z ⊂ X a subset with dimX Z � 0 and φ :X → 2Y an LSC map such that φ(x) is closed for all x /∈ C and Clos(φ(x))

is convex, for all x /∈ Z. Then φ has SEP.

Theorem 10. (See [18, Theorem 1.2].) Let X be a paracompact space, Y a Banach space, Z ⊂ X a subset withdimX Z � n+1 and φ :X →F(Y ) an LSC map such that and φ(x) is convex, for all x /∈ Z and the family {φ(x): x ∈Z} is uniformly equi-LCn. Then φ has SNEP. If moreover, φ(x) is n-connected for every x ∈ Z, then φ has SEP.

Note that in [18] the technique of the proof in [5] was rearranged in a more structured form, with exact extractingof the useful properties like SEP, SNEP and SAP (selection approximation property).

The joint paper with Mägerl and Mauldin [20] formally contains no “selections” in the title or in the statements ofthe main theorems (1.1 and 1.2). Nevertheless, the essence of these theorems is contained in the selection result.

It is a classical fact that each metric compact X can be represented as the image of the Cantor set K under somecontinuous surjection h :K → X. Theorem 5.1 of [20] states that if {Xα} is a family of subcompacta of a metricspace X which is continuously parameterized by α ∈ A with dimA = 0 then one can choose a family of surjectionshα :K → Xα continuously depending on the same parameter α ∈ A. Such parameterized version of the Alexandrovtheorem is in fact, derived from the Zero-dimensional selection theorem.

In general, the decade 1980–1990 was marked by Michael’s very diverse set of papers on selections, practicallyevery one of which contained new ideas of high quality. For one more example, the Finite-dimensional selectiontheorem from [5] was strengthened in [23] simultaneously in two directions. First, the assumption that {φ(x)}x∈X isan equi-LCn family in Y was replaced by the property that fibers {{x} × {φ(x)}x∈X} constitutes an equi-LCn familyin X × Y . This answered the problem of Eilenberg stated in 1956 (see the comments in [5]). Next, the closednessassumption for φ(x) ⊂ Y can be weakened to the closedness of graph-fibers {x} × φ(x) in some Gδ-subset of X × Y .

The key ingredient of the proof was a “factorization” construction. Briefly, it turned out that the LSC mappingφ :X → Y with weakened assumptions can be represented as a composition φ = h ◦ ψ with singlevalued h :Z → Y

and with ψ :X → Z, where ψ satisfies the classical assumptions of the Finite-dimensional selection theorem [5].Hence the composition of a selection of ψ with h gives the desired selection of φ.


We guess that the idea of the appearance of the Gδ-conditions was a corollary of constructions of selections,avoiding a countable set of obstructions, from the paper [22] which appeared one year earlier:

Theorem 11. (See [22, Theorem 3.3].) Let X be a paracompact space, Y a Banach space and φ :X → F(Y ) a LSCmap with convex values. Let ψi :X →F(Y ), i ∈ N, be continuous, Zi = {x ∈ X: φ(x)∩ψi(x) �= ∅} and suppose that

dimX < dimφ(x) − dim(conv

(φ(x) ∩ ψi(x)

)),

for all x ∈ Zi and i ∈ N. Then φ admits a selection f which avoids every ψi : f (x) /∈ ψi(x).

Briefly, in the values φ(x) there is sufficient “room” to avoid all sets ψi(x).Based on [22,23], Michael stated in “Open problems in topology, I” the “Gδ”-problem [24, Problem 396]: Does

the Convex-valued selection theorem remain true if φ maps X into some Gδ-subset Y of a Banach space B withconvex values which are closed in Y ? In spite of numerous cases with affirmative answer this problem has in generala negative (as it was convected in [24]) solution, for details see the paper of Namioka and Michael in this issue.

7. Papers from 1992

In general, all papers [21,25,27] are related to “dispersed”, mainly to zero-dimensional, (in dim-sense) domains ofmultivalued mappings.

Briefly, in [25] results of [17, Theorem 1.1] and [19, Theorem 1.3] are unified and generalized in the spirit of [23]to subsets C ⊂ X, which are unions of countable family of Gδ-subsets Cn of X and to a mappings φ, having SNEPat each Cn. In the paper written with Choban [27], the Compact-valued selection Theorem 6 was derived from theZero-dimensional one (Theorem 4). In fact, a paracompact domain X was represented as the image h(Z) of some zero-dimensional paracompact space Z with respect to some appropriate continuous (perfect or inductively open) mappingh : Z → X. Theorem 4 applied to the composition φ ◦ h gives a selection, say s :Z → Y . So, the composition s ◦ h−1

will be a desired multivalued selection of φ :X → F(Y ).The pair of papers [26,28] is related to “the differences between selection theorems which assume that the domain

is finite-dimensional and those which do not”. More generally, based on the Pixley counterexample in [26] it wasshown that a genuine dimension-free analogue of the Finite dimensional selection theorem does not exist or briefly,that there are no purely topological analogs of convexity. In comparison, in [28] a convexity, or connectivity typerestrictions in the spirit of [10] for a mapping are presented and under such restrictions the equivalence is provedbetween the existence of global selections and the existence of selections locally.

The paper [29] on continuum-valued selections is an elegant simultaneous application of the “universality” ideafrom [27] and the one-dimensional selection theorem (special case n = 0 of Theorem 5). The key step can be describedas follows. Due to a recent theorem of Pasynkov, each paracompact domain X can be represented in the form h(Z),for some perfect, open surjection h :Z → X with pathwise connected fibers and for some paracompact space Z withdimZ � 1. So, if the composition φ ◦ h admits a selection, say s :Z → Y then the composition s ◦ h−1 will be acontinuum-valued selection of φ :X →F(Y ).

We should mention the thoughtfulness, exactness and perfectness of all Michael’s papers. His laconic style ofexposition is perfectly matched with the deepness of his results. In our opinion, A man of few words but with greatideas could well serve as a good description of his character. As a rule, all his papers are equipped with a considerablenumber of additional references, which were added at proofs, and which very precisely give correct accents to thepaper needed for proper understanding. In conclusion of this survey of Michael’s results on selections we wish ourjubilant successful realization of many more projects.

Acknowledgements

We thank Jan van Mill for comments and suggestions. The first author was supported by the Slovenian ResearchAgency grants No. P1-0292-0101-04 and Bl-RU/05-07-04. The second author was supported by the RFBR grantNo. 05-01-00993.


References

[R1] J.-P. Aubin, A. Cellina, Differential Inclusions, Set-Valued Maps and Viability Theory, Grundl. der Math. Wiss., vol. 264, Springer-Verlag,Berlin, 1984.

[R2] J.-P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Basel, 1990.[R3] G. Beer, Topologies on Closed and Convex Sets, Kluwer, Dordrecht, 1993.[R4] C. Bessaga, A. Pelczynski, Selected Topics in Infinite-dimensional Topology, Monogr. Math., vol. 58, PWN, Warszaw, 1975.[R5] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis, V.V. Obuhovskij, Set-valued maps, Itogi Nauki Tehn. Mat. Anal. 19 (1993) 127–230 (in

Russian).[R6] F. Deutsch, A survey of metric selections, Contemp. Math. 18 (1983) 49–71.[R7] J. Dugundji, A. Granas, Fixed Point Theory, Monogr. Math., vol. 61, PWN, Warsaw, 1982.[R8] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications, vol. 495, Kluwer, Dordrecht,

1999.[R9] J. van Mill, Infinite-Dimensional Topology: Prerequisites and Introduction, North-Holland, Amsterdam, 1989.[R10] J. van Mill, The Infinite-Dimensional Topology of Function Spaces, North-Holland, Amsterdam, 2001.[R11] J.-I. Nagata, Modern General Topology, second ed., North-Holland Math. Libr., vol. 3, North-Holland, Amsterdam, 1985.[R12] T.P. Parthasaraty, Selection Theorems and their Applications, Lecture Notes Math., vol. 263, Springer-Verlag, Berlin, 1972.[R13] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, Mathematics its Applications, vol. 455, Kluwer, Dordrecht,

1998.[R14] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, in: M. Hušek, J. van Mill (Eds.), Recent Progress in General

Topology II, Elsevier, Amsterdam, 2002, pp. 423–461.[R15] M.L.J. van de Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993.



Hyperspace of convex compacta of nonmetrizable compact convexsubspaces of locally convex spaces ✩

Lidia Bazylevych a, Dušan Repovš b,∗, Michael Zarichnyi c,d

a National University “Lviv Polytechnica”, 12 Bandery Str., 79013 Lviv, Ukraineb Institute of Mathematics, Physics and Mechanics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana, Slovenia 1001

c Department of Mechanics and Mathematics, Lviv National University, Universytetska Str. 1, 79000 Lviv, Ukrained Institute of Mathematics, University of Rzeszów, Rejtana 16 A, 35-310 Rzeszów, Poland

Received 6 November 2006; accepted 23 February 2007

Abstract

Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space isan absolute retract if and only if X is an absolute retract of weight � ω1. It is also proved that the hyperspace of convex compactsubsets of the Tychonov cube Iω1 is homeomorphic to Iω1 . An analogous result is also proved for the cone over Iω1 . Our proofs arebased on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.© 2007 Elsevier B.V. All rights reserved.

MSC: 54B20; 54C55; 46A55

Keywords: Compact convex set; Hyperspace; Tychonov cube; Soft map

1. Introduction

For any uncountable cardinal number τ , the Tychonov and the Cantor cubes (denoted by I τ and Dτ , respectively),belong to the class of main geometric objects in the topology of non-metrizable compact Hausdorff spaces. The spacesI τ (we denote by I the segment [0,1]) and Dτ were first characterized by Shchepin [13]. In particular, the Tychonovcubes are characterized as the homogeneous-by-character nonmetrizable compact Hausdorff absolute retracts [12].This characterization was later applied to the study of topology of the functor-powers, i.e. spaces of the form F(Kτ ),where K is a compact metrizable space and F is a covariant functor in the category of compact Hausdorff spaces.In particular, it was proved that, for an uncountable τ , the space P(I τ ), where P denotes the probability measurefunctor, is homeomorphic to I τ if and only if τ = ω1. For the hyperspace functor exp it is known that exp(Dτ ) ishomeomorphic to Dτ if and only if τ = ω1 and exp(I τ ) is not an absolute retract whenever τ > ω.

In this paper we consider the hyperspaces cc(X) of nonempty compact convex subsets in X, for compact convexsubsets in locally convex spaces X. For metrizable X, this object was investigated by different authors (see, e.g.,

✩ The research was supported by the Slovenian–Ukrainian grant BI-UA/04-06-007. The authors are indebted to the referee for valuable remarks.* Corresponding author.

E-mail addresses: [email protected] (L. Bazylevych), [email protected] (D. Repovš), [email protected] (M. Zarichnyi).


L. Bazylevych et al. / Topology and its Applications 155 (2008) 764–772 765

[7,9]). In particular, it was proved in [9] that the hyperspace of convex compact subsets of the Hilbert cube Q = Iω ishomeomorphic to Iω.

The aim of this paper is to consider the nonmetrizable compact convex subsets in locally convex spaces. One of ourmain results is Theorem 4.1, which characterizes the compact convex spaces X by cc(X) being an absolute retract. Wealso show that the space cc(X) is homeomorphic to Iω1 (resp., the cone over Iω1 ) if and only if X is homeomorphicto Iω1 (resp., the cone over Iω1 ).

These results are in the spirit of the corresponding results concerning the functor-powers of compact metric spaces(see [13]). The proofs are based on the spectral analysis of nonmetrizable compact Hausdorff spaces, in particularon the Shchepin Spectral Theorem [13] as well as on analysis of the selection type properties of the maps of thehyperspaces of compact convex subsets.

The construction cc determines a functor acting on the category Conv of compact convex subsets of locally convexspaces. The results of this paper demonstrate that the functor cc is closer to the functor P of probability measures thanto the hyperspace functor exp.

2. Preliminaries

All topological spaces are assumed to be Tychonov, all maps are continuous. By A we denote the closure of asubset A of a topological space. Let X be any space.

The hyperspace expX of X is the space of all nonempty compact subsets in X endowed with the Vietoris topology.A base of this topology is formed by the sets of the form

〈U1, . . . ,Un〉 = {A ∈ expX | A ⊂ U1 ∪ · · · ∪ Un and A ∩ Ui = ∅ for every i},where U1, . . . ,Un run through the topology of X, n ∈ N. For a metric space (X,ρ), the Vietoris topology on exp(X)

is induced by the Hausdorff metric ρH:

ρH(A,B) = inf{ε > 0 | A ⊂ Oε(B), B ⊂ Oε(A)

}.

The hyperspace construction determines a functor in the category Comp of compact Hausdorff spaces and con-tinuous maps. Given a map f :X → Y in Comp, we define exp(f ) : exp(X) → exp(Y ) by exp(f )(A) = f (A),A ∈ exp(X).

Let Conv denote the category of compact convex subsets in locally convex spaces and affine continuous maps. IfX is an object of Conv we define

cc(X) = {A ∈ exp(X) | A is convex

} ⊂ exp(X).

If f :X → Y is a map in Conv, then the map cc(f ) : cc(X) → cc(Y ) is defined as the restriction of exp(f ) on cc(X).In the sequel, for a nonempty compact subset X in a locally convex space Y , we denote the closed convex hull map

by h : expX → cc(Y ). Let X be a subset of a metrizable locally convex space M . In the sequel, we identify any pointx ∈ X with the singleton {x} ∈ cc(X).

Recall that the Minkowski operation in cc(X) is defined as follows:

λ1A1 + λ2A2 = {λ1x1 + λ2x2 | x1 ∈ A1, x2 ∈ A2},λ1, λ2 ∈ R, A1,A2 ∈ cc(X).

Lemma 2.1. Let X be a compact convex subset in a locally convex space. There exists an embedding α of the spacecc(X) into a locally convex space L satisfying the condition

α(λ1A1 + λ2A2) = λ1α(A1) + λ2α(A2) (2.1)

for every λ1, λ2 ∈ R, A1,A2 ∈ cc(X).

Proof. Let X be a compact convex subset in a metrizable locally convex space M . Following [11], consider theequivalence relation ∼ on cc(M) × cc(M) defined by the condition: (A,B) ∼ (C,D) if and only if A + D = B + C.Denote by L the set of equivalence classes of ∼ (in the sequel, we denote by [A,B] the equivalence class that contains

766 L. Bazylevych et al. / Topology and its Applications 155 (2008) 764–772

(A,B)). It is well known that L is a linear space with respect to the naturally defined operations. Let U be a convexneighborhood of zero in M and define

U∗ = {[A,B] ∈ L | A ⊂ B + U, B ⊂ A + U}.

The sets U∗ form a base at zero in L. The map α : cc(X) → L defined by the formula α(A) = [A, {0}] is the requiredembedding. �3. Functor cc and soft maps

A map f :X → Y is soft (see [13]) if for every commutative diagram

Aψ

i

X

f

Z ϕ Y

where i :A → Z is a closed embedding into a paracompact space Z, there exists a map Φ :Z → X such that Φ|A = ψ

and f Φ = ϕ.In other words, a map is soft if it satisfies the parameterized selection extension property.The following proposition is close to the Michael selection theorem for convex-valued maps [8].

Proposition 3.1. Let f :X → Y be an affine open map of compact convex metrizable subsets of locally convex spaces.Then the map cc(f ) : cc(X) → cc(Y ) is soft.

Proof. We first prove that the map cc(f ) is open. It is well known that the map exp(f ) is open. Since the diagram

(exp(f ))−1(cc(Y ))h

exp(f )|(exp(f ))−1(cc(Y ))

cc(X)

cc(f )

cc(Y )

is commutative and the closed convex hull map h is a retraction of (exp(f ))−1(cc(Y )) onto cc(X), we see that the mapcc(f ) is also open.

There exists an embedding α : cc(X) → L satisfying condition (2.1). Choose a countable family of functionals{ϕ1, ϕ2, . . .} ⊂ L∗ such that this family separates the points and ϕi(α(cc(X))) ⊂ [−1/i,1/i]. Then the map ϕ =(ϕ1, ϕ2, . . .), defined on α(cc(X)), embeds α(cc(X)) into the Hilbert space �2. Denote by

ξ :ϕ(α(cc(X)

)) × cc(ϕ(α(cc(X))

)) → ϕ(α(cc(X)

))

the nearest point map: y = ξ(x,A) if and only if ‖z− x‖ > ‖y − x‖, for every z ∈ A \ {y} (here ‖ · ‖ denotes the normin �2).

Suppose a commutative diagram

Ap

cc(X)

cc(f )

Z q cc(Y )

is given, where A is a closed subset of a paracompact space Z.Since cc(X) is an absolute retract, there exists a map r :Z → cc(X) such that r|A = p. Note that for every

B ∈ cc(Y ), the set ϕ(α(cc(f )−1(B))) is a convex closed subset of ϕ(α(cc(X))), i.e. an element of the spacecc(ϕ(α(cc(X)))). Since the map cc(f ) is open, the map

δ : cc(Y ) → cc(ϕ(α(cc(X))

)), δ(B) = ϕ

(α(cc(f )−1(B)

)),

is continuous.


Define the map R :Z → cc(X) by the formula

R(z) = α−1(ϕ−1(ξ(ϕ(α(r(z))), δ(q(z)))))

, z ∈ Z.

It is easy to see that R is continuous, R|A = p, and cc(f )R = q . �A point p of a set X in a locally convex space E is called an exposed point of X if there exists a continuous linear

functional f on E such that f (x) > f (p), for each x ∈ X \ {p}.

Lemma 3.2. Let f :X → Y be an open affine continuous map of compact convex subsets in locally convex spacessuch that |f −1(y)| > 1 for every y ∈ Y . Then |cc(f )−1(B)| > 1, for every B ∈ cc(Y ).

Proof. As in the proof of Proposition 3.1, one may assume that X is affinely embedded in the Hilbert space �2. LetB ∈ cc(Y ) and A ∈ cc(f )−1(B). If A = f −1(B), then we define A′ as the closure of the convex hull of A∪{x}, wherex ∈ f −1(B) \ A. Then A′ = A and A′ ∈ cc(f )−1(B).

If A = f −1(B), then it is well-known (see e.g., [1]) that there exists an exposed point, x of A. Since f is open,there exists a neighborhood U of x such that f (A \ U) = B . In this case we define A′ as the closure of the convexhull of A \ U . Note that A′ ∈ cc(f )−1(B). That A = A′ easily follows from the fact that x is an exposed point. �Lemma 3.3. Suppose that f :X → Y is a continuous affine map of compact convex subsets of locally convex spaces.If the map cc(f ) is open then so is the map f .

Proof. Suppose to the contrary, that f is not open. Then there exists x ∈ X and a net (yα)α∈A in Y converging toy = f (x), such that there is no net (xα)α∈A in X converging to x with xα ∈ f −1(yα), for every α ∈ A.

Assuming that the map cc(f ) is open, we obtain that there exists a net (Cα)α∈A in cc(X) converging to {x} and suchthat cc(f )(Cα) = {yα}, for every α ∈ A. Then, obviously, the net (cα)α∈A converges to x, for every choice cα ∈ Cα ,α ∈ A. This gives a contradiction. �

A commutative diagram

D = Xf

g

Y

u

Z v T

(3.1)

is called soft if its characteristic map

χD = (f, g) :X → Y ×T Z = {(y, z) ∈ Y × Z | u(y) = v(z)

}

is soft.

Lemma 3.4. Suppose that a commutative diagram D (see formula (3.1)) in the category Conv consists of metrizablespaces. If the diagram cc(D) is soft, then so is the diagram D.

Proof. First we show that the diagram D is open if such is cc(D). Let (yi, zi)∞i=1 be a sequence in Y ×T Z converging

to a point (y, z) and let x ∈ X be such that χD(x) = (y, z). Since cc(D) is soft (and therefore open), there exists asequence (Ai)

∞i=1 in cc(X) such that (f (Ai), g(Ai)) = ({yi}, {zi}), for every i, and (Ai)

∞i=1 converges to {x} in cc(X).

Choose arbitrary xi ∈ Ai , then (f (xi), g(xi)) = (yi, zi), for every i, and (xi)∞i=1 converges to x in X. This shows that

the map χD is open.Now the map χD , being an open affine map of convex compact metrizable subspaces of locally convex spaces, is

soft. This follows from the Michael Selection Theorem [8] (see e.g., [13]). �4. Hyperspaces cc(X) homeomorphic to Tychonov cubes

We are going to recall some definitions and results related to the Shchepin Spectral Theorem (see [13] for details).In what follows, an inverse system S = {Xα,pαβ;A} satisfies the following conditions:


(1) Xα are compact Hausdorff spaces;(2) pαβ are surjective;(3) the partially ordered set A (by �) is directed, i.e., for every α,β ∈ A there exists γ ∈ A with α � γ , β � γ .

An inverse system S = {Xα,pαβ;A} is called open if all the maps pαβ are open. An inverse system S ={Xα,pαβ;A} is called continuous if for every α ∈A we have Xα = lim←−{Xα′ ,pα′β ′ ;α′, β ′ < α}.

By w(X) we denote the weight of a space X. An inverse system S = {Xα,pαβ;A} is called a τ -system, τ beinga cardinal number, if the following holds:

(1) the directed set A is τ -complete, i.e. every chain of cardinality � τ in A has the least upper bound;(2) S is continuous;(3) w(Xα) � τ , for every α ∈A.

If τ = ω, we use the terms σ -complete and σ -system.For every A, we denote the family of all countable subsets of A ordered by inclusion by Pω(A).A standard way to represent a compact Hausdorff space X as a limit of a σ -system is to embed it into a Tychonov

cube I τ , for some τ . For any countable A ⊂ τ , let XA = pA(X), where pA : I τ → IA denotes the projection. In thisway we obtain an inverse system S = {XA,pAB;Pω(τ)}, where, for A ⊃ B , pAB :XA → XB denotes the (unique)map with the property pB |X = pAB(pA|X). The resulting inverse system S is a σ -system and X = lim←−S .

If X is a compact convex subset of a locally convex space, we can affinely embed X into I τ , for some τ . The aboveconstruction gives us an inverse σ -system S in the category Conv such that X = lim←−S .

In the sequel, we will use the well-known fact that the functor cc is continuous in the sense that it commutes withthe limits of inverse systems.

A compact Hausdorff space X is openly generated if X is the limit of an inverse σ -system with open short projec-tions. The absolute retracts (ARs) are considered in the class of compact Hausdorff spaces.

Theorem 4.1. Let X be a convex compact subset of a locally convex space. Then the space cc(X) is an absolute retractif and only if X is openly generated and of weight � ω1.

Proof. If X is openly generated and of weight � ω1, then X is homeomorphic to lim←−S , where S = {Xα,pαβ;ω1}is an inverse system consisting of convex compact subsets of metrizable locally convex spaces and open maps. Thencc(X) is homeomorphic to lim←− cc(S). Since the spaces cc(Xα) are ARs and the maps cc(pαβ) are soft (see Proposition

3.1), the space cc(X) is an AR.Suppose now that cc(X) is an AR of weight � ω2. It easily follows from standard results of Shchepin’s theory

that there exists a compact convex space X of weight ω2 such that cc(X) is an AR (see [13] and also [4], where thecase of locally convex spaces is considered). We may assume that cc(X) = lim←− cc(S), where S = {Xα, pαβ;ω2} is

an inverse system such that for every α < ω2 the space cc(Xα) is an AR and for every α,β , β � α < ω2, the mapcc(pαβ) is soft. In its turn, every Xα can be represented as lim←− Sα , where Sα = {Xαγ , qα

γ δ;ω1} is an inverse system

in Conv and it follows from the results of Chigogidze [4] that for every α,β , where β � α < ω2, the map pαβ is thelimit of a morphism (p

γαβ)γ<ω1 : Sα → Sβ such that the maps cc(pγ

αβ) are soft and for every γ � δ, γ, δ < ω1, thediagram

cc(Xαγ )cc(pγ

αβ )

cc(qαγ δ)

cc(Xβγ )

cc(qβγ δ)

cc(Xαδ) cc(pδαβ )

cc(Xβδ)

is soft. Since all the spaces in the above diagram are metrizable, by Lemma 3.4, the diagram


Xαγ

pγαβ

qαγ δ

Xβγ

qβγ δ

Xαδ pδαβ

Xβδ

is also soft. As the limits of soft morphisms, the maps pαβ are soft and we conclude that the space X is an absoluteretract.

Since the space X is an AR, it contains a copy of the Tychonov cube Iω2 . It follows from the Shchepin SpectralTheorem that, without loss of generality, one may assume that every Xα contains the space (Iω1)α and for every α,β ,where β � α < ω2, the map pαβ |(Iω1)α is the projection map of (Iω1)α onto (Iω1)β .

Denote by D the Aleksandrov supersequence of weight ω1, i.e. the one-point compactification of a discrete spaceof cardinality ω1.

Claim. There exists α < ω2 such that the subspace (Iω1)α ⊂ Xα contains an affinely independent copy of thespace D.

Proof. Represent D as {dγ | γ � ω1}, where dω1 denotes the unique non-isolated point of D. For γ < ω1, let rγ :D →{dδ | δ � γ } ∪ {dω1} denote the retraction that sends {dδ | γ < δ < ω1} into dω1 .

Define by transfinite induction maps fγ :D → (Iω1)αγ ⊂ Xαγ , where γ < ω1 and αγ < ω2, so that αγ � αγ ′ andpαγ ′αγ fγ ′ = fγ for every γ � γ ′.

Let f0 :D → (Iω1)α0 ⊂ Xα0 be an arbitrary constant map, for some α0 < ω2. Suppose that, for some δ < ω1,maps fγ are already defined for every γ < δ so that fγ = iγ rγ for some embedding iγ : rγ (D) → Xαγ . If δ is a limitordinal, let αδ = sup{αγ | γ < δ} and fδ = lim←−{fγ | γ < δ}. If δ = δ′ + 1, let αδ = αδ′ + 1 and find an embedding

iδ : rδ(D) → (Iω1)αδ ⊂ Xαδ such that pαδαδ′ iδ = iδ′ and pαδαδ′ iδ(dδ) = iδ′(dδ′). Put fδ = iδrδ .Finally, let α = sup{αγ | γ < ω1} and f = lim←−{fγ | γ < ω1}. Claim is thus proved. �We now return to the proof of the theorem. Without loss of generality, we assume that D ⊂ (Iω1)α ⊂ Xα and D is

affinely independent in Xα . Recall that h(D) denotes the closed convex hull of D in Xα . We are going to show that thespace (cc(pα+1,α))−1(h(D)) does not satisfy the Souslin condition. There exist two maps s1, s2 :D → Xα+1 such thatpα+1,αs1 = pα+1,αs2 = 1D and s1(D) ∩ s2(D) = ∅. Let U1,U2 be neighborhoods of s1(D) and s2(D), respectively,such that U1 ∩ U2 = ∅.

For every isolated point y ∈ D let Vy be a neighborhood of y in Xα such that Vy ∩ h(D \ {y}) = ∅.Let

Wy = ⟨Xα+1 \ (

U2 ∩ p−1α+1,α

(D \ {y})),U2 ∩ p−1

α+1,α(Vy)⟩.

We are going to show that cc(pα+1,α)−1(h(D)) ∩ Wy = ∅. To this end, consider the set B = h(s1(D \ {y}) ∪{s2(y)}). Obviously, B ∈ cc(pα+1,α)−1(h(D)) and s2(y) ∈ B ∩U2 ∩ p−1

α+1,α(Vy). In addition, for every z ∈ D \ {dω1},z = y, we have B ∩ p−1

α+1,α(z) = {s1(z)}, therefore B ⊂ Xα+1 \ (U2 ∩ p−1α+1,α(D \ {y})). We conclude that B ∈ Wy .

It remains to prove that for every y, z ∈ D \ {dω1}, y = z, we have Wy ∩ Wz ∩ cc(pα+1,α)−1(h(D)) = ∅. Indeed,otherwise, for any A ∈ Wy ∩ Wz ∩ cc(pα+1,α)−1(h(D)) we would have A ∩ p−1

α+1,α(y) ⊂ p−1α+1,α(y) \ U2 and, on the

other hand, A ∩ p−1α+1,α(y) ⊂ U2, a contradiction. We therefore conclude that

{Wy ∩ cc(pα+1,α)−1(h(D)

) | y ∈ D \ {dω1}}

is a family of nonempty disjoint open subsets in cc(pα+1,α)−1(h(D)). Since the space cc(pα+1,α)−1(h(D)) does notsatisfy the Souslin condition, we obtain that cc(pα+1,α)−1(h(D)) /∈ AR and hence the map cc(pα+1,α) is not a softmap. This contradiction demonstrates that w(X) � ω1.

We are going to show that X is openly generated. Since cc(X) is an AR of weight ω1, there exists an inverse systemS = {Xα,pαβ;ω1} consisting of compact metrizable convex spaces and affine maps such that cc(X) = lim cc(S).
←−


Applying Shchepin’s Spectral Theorem, we may additionally assume that all the maps cc(pαβ), β � α < ω1, are soft.

By Lemma 3.3, the maps pαβ , β � α < ω1, are soft and therefore open. �Theorem 4.2. Let X be a convex compact subset of a locally convex space. The space cc(X) is homeomorphic to Iω1

if and only if X is homeomorphic to Iω1 .

Proof. We use the following characterization of the Tychonov cube I τ , τ > ω, due to Shchepin [13]: a compactHausdorff space X of weight τ > ω is homeomorphic to the Tychonov cube I τ if and only if X is a characterhomogeneous absolute retract. Recall that a space is called character homogeneous if the characters of all of itspoints are equal.

If the weight of X is ω1, then it easily follows from the Shchepin Spectral Theorem [13] that X can be representedas lim←−S , where S = {Xα,pαβ;ω1} is an inverse system consisting of convex compact metrizable subsets in locally

convex spaces and affine continuous maps. Since the functor cc is continuous (see, e.g., [10]), we obtain that cc(X) =lim←−{cc(Xα), cc(pαβ);ω1}. Since cc(Xα) is an absolute retract (see [15]) and, by Proposition 3.1, the map cc(pαβ) is

soft for every α,β < ω1, α � β , we apply a result of Shchepin (see [13]) to derive that cc(X) is an absolute retract.If X is character homogeneous, then we can in addition assume that no projection pαβ possesses one-point

preimages. By Lemma 3.2, the maps cc(pαβ) do not possess one-point preimages and therefore cc(X) is characterhomogeneous. By the mentioned result of Shchepin, cc(X) is homeomorphic to Iω1 .

If cc(X) is homeomorphic to Iω1 , then there exists an inverse system S = {Xα,pαβ;ω1} consisting of compactmetrizable convex spaces and open affine maps such that cc(X) = lim←− cc(S). Applying Shchepin’s Spectral Theorem,

we may additionally assume that all the maps cc(pαβ), β � α < ω1, are soft and do not possess points with one-pointpreimage. It is then evident that the maps pαβ , β � α < ω1, do not possess points with one-point preimage. ApplyingLemma 3.3 we conclude that the maps pαβ , β � α < ω1, are open and therefore, by the Michael Selection Theorem,soft. Then X is a character homogeneous AR of weight ω1. By the cited characterization theorem for Iω1 , the spaceX is homeomorphic to Iω1 . �5. Cone over Tychonov cube

Define the cone functor cone in the category Conv as follows. Given an object X in Conv, i.e. a compact convexsubset X in a locally convex space L, let cone(X) be the convex hull of the set X × {0} ∪ {(0,1)} in L × R. For amorphism f :X → Y in Conv define cone(f ) : cone(X) → cone(Y ) as the only affine continuous map that extendsf × {0} :X × {0} → Y × {0} and sends (0,1) ∈ cone(X) to (0,1) ∈ cone(Y ).

We will need the following notion. A map f :X → Y is called a trivial Q-bundle if there exists a homeomorphismg :X → Y × Q such that f = pr1 g. The following statement is a characterization theorem for the space cone(Iω1)

among the convex compact spaces.

Proposition 5.1. A convex compactum X is homeomorphic to the space cone(Iω1) if and only if X satisfies thefollowing properties:

(1) X is an AR;(2) w(X) = ω1; and(3) there exists a unique point x ∈ X of countable character.

Proof. Obviously, if a convex compactum X is homeomorphic to cone(Iω1), then X satisfies properties (1)–(3).Suppose now that X satisfies (1)–(3). Then X is homeomorphic to the limit of a continuous inverse system S =

{Xα,pαβ;ω1} in Conv which satisfies the properties

(i) Xα is a convex metrizable compactum for every α;(ii) pαβ is an open affine map for every α � β; and

(iii) {xβ} = {y ∈ Xβ | |p−1αβ (y)| = 1}.


Passing, if necessary, to a subsystem of S , one can assume that for every α and every compact subset K of Xα \ {xα}the map

pα+1,α|p−1α+1,α(K) :p−1

α+1,α(K) → K

satisfies the condition of fibrewise disjoint approximation. The Torunczyk–West characterization theorem [14] impliesthat, if K is an AR, the map pα+1,α|p−1

α+1,α(K) is a trivial Q-bundle and therefore the map

pα+1,α|p−1α+1,α

(Xα \ {xα}) = pα+1,α|(Xα+1 \ {xα+1}

),

being a locally trivial Q-bundle, is a trivial Q-bundle (see [2]). Therefore, the map pα+1,α is homeomorphic to theprojection map pr23 :Q × Q × [0,1) → Q × [0,1) (that Xα \ {xα} is homeomorphic to Q × [0,1) follows from thefact that the spaces Q and cone(Q) are homeomorphic—see [3]). Passing to the one-point compactifications of thesemaps we obtain the commutative diagram

Xα+1

pα+1,α

cone(Q × Q)

cone(pr2)

Xα+1 \ {xα+1}pα+1,α |...

Q × Q × [0,1)

pr23

Xα \ {xα} Q × [0,1)

Xα cone(Q)

in which the horizontal arrows are homeomorphisms. Therefore X and cone(Iω1) are homeomorphic. �Theorem 5.2. Let X be an object of the category Conv. The space cc(X) is homeomorphic to the cone over theTychonov cube, cone(Iω1), if and only if X is homeomorphic to the space cone(Iω1).

Proof. Suppose that a convex compact space X is an absolute retract of weight ω1 with exactly one point x, ofcountable character. It follows from the Shchepin Spectral Theorem ([13]; see also [4]) that X can be represented aslim←−S , where S = {Xα,pαβ;ω1} is an inverse system in which every Xα is a metrizable convex compactum and everypαβ , α � β , is an affine map. Denote by pα :X → Xα the limit projections and let xα = pα(x). Passing, if necessary,to a subsystem of S , one can assume additionally that for every α � β we have {xβ} = {y ∈ Xβ | |p−1

αβ (y)| = 1}.Then for every α � β , the map cc(pαβ) is a soft map and similarly as in the proof of Lemma 3.2, one can show that

{{xβ}} = {A ∈ cc(Xβ) | ∣∣cc(pαβ)−1(A)

∣∣ = 1}.

We conclude that the space cc(X) = lim←−(S) satisfies the conditions of Proposition 5.1 and therefore is homeomorphic

to the space cone(Iω1).Now, if cc(X) is homeomorphic to cone(Iω1), it follows from Theorem 4.1 that X is an AR of weight ω1. Note that

for every point x of countable character in X, the point {x} is of countable character in cc(X). We therefore concludethat there is a unique point of countable character in X. By Proposition 5.1, X is homeomorphic to cone(Iω1). �6. Remarks and open problems

Problem 6.1. Let f :X → Y be an affine continuous map of compact metrizable compacta in locally convex spacessuch that dimf −1(y) � 2, for every y ∈ Y . Is the map cc(f ) : cc(X) → cc(Y ) homeomorphic to the projection mappr1 :Q × Q → Q?

Note that there is an open map f :X → Y of metrizable compacta with infinite fibers such that the mapP(f ) :P(X) → P(Y ) is not homeomorphic to the projection map pr1 :Q × Q → Q (see [6]). (Recall that P de-notes the probability measure functor.)


Problem 6.2. Does every compact convex AR of weight τ � ω1 contain an affine copy of the Tychonov cube I τ ?

It is known that every compact Hausdorff AR of weight τ � ω1 contains a topological copy of the Tychonov cubeI τ (see [12]).

The theory of nonmetrizable noncompact absolute extensors which is, in some sense, parallel to that of compactabsolute extensors, was elaborated by Chigogidze [4,5]. One can also consider the hyperspaces of compact subsets inthe spaces R

τ and conjecture that for noncountable τ , the hyperspace cc(Rτ ) is homeomorphic to Rτ if and only if

τ = ω1.

References

[1] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc., Provi-dence, RI, 2000.

[2] T.A. Chapman, Locally-trivial bundles and microbundles with infinite-dimensional fibers, Proc. Amer. Math. Soc. 37 (2) (1973) 595–602.[3] T.A. Chapman, Lectures on Hilbert Cube Manifolds, Conf. Board Math. Sci. Reg. Conf. Ser. Math., vol. 28, Amer. Math. Soc., Providence,

RI, 1975.[4] A. Chigogidze, Locally convex linear topological spaces that are homeomorphic to the powers of the real line, Proc. Amer. Math. Soc. 113

(1991) 599–609.[5] A. Chigogidze, Nonmetrizable ANR’s admitting a group structure are manifolds, Topology Appl. 153 (7) (2006) 1079–1083.[6] A.N. Dranishnikov, Q-bundles without disjoint sections, Funct. Anal. Appl. 22 (2) (1988) 151–152.[7] L. Montejano, The hyperspace of compact convex subsets of an open subset of R

n, Bull. Pol. Acad. Sci. Math. 35 (11-12) (1987) 793–799.[8] E. Michael, Continuous selections I, Ann. of Math. 63 (2) (1956) 361–382.[9] S.B. Nadler Jr., J. Quinn, N.M. Stavrakos, Hyperspace of compact convex sets, Pacific J. Math. 83 (1979) 441–462.

[10] O.R. Nykyforchyn, On homeomorphisms of hyperspaces of convex subsets, Mat. Stud. 5 (1995) 57–64.[11] A.G. Pinsker, Space of convex sets of locally convex space, Trudy Leningr. Inzh.-Ekon. Inst. 63 (1966) 13–17.[12] E.V. Shchepin, On Tychonov manifolds, Sov. Math., Dokl. 20 (1979) 511–515.[13] E.V. Shchepin, Functors and uncountable powers of compacta, Uspekhi Mat. Nauk 31 (1981) 3–62.[14] H. Torunczyk, J. West, Fibrations and Bundles With Hilbert Cube Manifold Fibers, Amer. Math. Soc. Memoirs, vol. 406, Amer. Math. Soc.,

Providence, RI, 1989.[15] M. Zarichnyi, S. Ivanov, Hyperspaces of compact convex subsets in the Tychonov cube, Ukrainian Math. J. 53 (5) (2001) 809–813.



Sections of Serre fibrations with 2-manifold fibers

N. Brodsky a,∗, A. Chigogidze b, E.V. Šcepin c

a University of Tennessee, Knoxville, TN 37996, USAb University of North Carolina at Greensboro, Greensboro, NC 27402, USA

c Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow 117966, Russia

Received 3 June 2006; received in revised form 19 September 2006; accepted 19 September 2006

Abstract

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided everyfiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney’s theorem to the case when all fibers arehomeomorphic to a given compact two-dimensional manifold.© 2007 Published by Elsevier B.V.

MSC: primary 57N05, 57N10; secondary 54C65

Keywords: Serre fibration; Section; Selection; Approximation

1. Introduction

The following problem is one of the central problems in geometric topology [7]. Let p :E → B be a Serre fibrationof separable metric spaces. Suppose that the space B is locally n-connected and all fibers of p are homeomorphic toa given n-dimensional manifold Mn. Is p a locally trivial fibration?

For n = 1 an affirmative answer to this problem follows from results of H. Whitney [19].

Conjecture (Šcepin). A Serre fibration with a locally arcwise connected metric base is locally trivial if every fiber ofthis fibration is homeomorphic to a given manifold Mn of dimension n � 4.

In dimension n = 1 Šcepin’s Conjecture is valid even for non-compact fibers [15]. Šcepin also proved that thepositive solution of the Conjecture in dimension n implies positive solutions of both the CE-problem and the Homeo-morphism Group problem in dimension n [16,7]. Since the CE-problem was solved in negative by A.N. Dranishnikov,it follows that there are dimensional restrictions in Šcepin’s Conjecture.

The first step toward proving Šcepin’s Conjecture in dimension n = 2 was made in [3], where existence of localsections of the fibration was proved under the assumption that the base space is an ANR. We improve this result in two

* Corresponding author.E-mail addresses: [email protected] (N. Brodsky), [email protected] (A. Chigogidze), [email protected] (E.V. Šcepin).

0166-8641/$ – see front matter © 2007 Published by Elsevier B.V.doi:10.1016/j.topol.2006.09.019

774 N. Brodsky et al. / Topology and its Applications 155 (2008) 773–782

directions. First we prove that any section of the fibration over closed subset A of the base space can be extended to asection over some neighborhood of A. Secondly, we prove a theorem on global sections of the fibration.

Theorem 4.4. Let p :E → B be a Serre fibration of locally connected compacta with all fibers homeomorphic togiven two-dimensional manifold. If B ∈ ANR, then any section of p over a closed subset A ⊂ B can be extended to asection of p over a neighborhood of A.

Theorem 4.9. Let p :E → B be a Serre fibration of locally connected compactum E onto an ANR-compactum B

with all fibers homeomorphic to given two-dimensional manifold M . If M is not homeomorphic to the sphere or theprojective plane, then p admits a global section provided one of the following conditions holds:

(a) π1(M) is abelian and H 2(B;π1(Fb)) = 0.(b) π1(M) is non-abelian, M is not homeomorphic to the Klein bottle and π1(B) = 0.(c) M is homeomorphic to the Klein bottle and π1(B) = π2(B) = 0.

Our strategy of constructing a section of a Serre fibration is as follows. We consider the inverse (multivalued)mapping and find its compact submapping admitting continuous approximations. Then we take very close continuousapproximation and use it to find again a compact submapping with small diameters of fibers admitting continuousapproximations. When we continue this process we get a sequence of compact submappings with diameters of fiberstending to zero. This sequence will converge to the desired singlevalued submapping (selection).

Section 3 is devoted to continuous approximations of multivalued maps. We prove filtered finite dimensional ap-proximation theorem (Theorem 3.13) and then apply it in a usual way (compare with [9]) to prove an approximationtheorem for maps of ANR-spaces. Since we are going to use singular filtrations of multivalued maps instead of usualfiltrations, our Theorem 3.13 generalizes the Filtered Approximation Theorem proved in [17]. But the proof of oursingular version of Filtered Approximation Theorem in full generality requires a lot of technical details. Consequentlywe decided to present only the version that we need—for compact maps of metric spaces.

Let us recall some definitions and introduce our notation. All spaces will be separable metrizable. By a mappingwe understand a continuous single-valued mapping. We equip the product X × Y with the metric

distX×Y

((x, y), (x′, y′)

) = distX(x, x′) + distY (y, y′).By O(x, ε) we denote the open ε-neighborhood of the point x.

A multivalued mapping F :X → Y is called a submapping (or selection) of a multivalued mapping G :X → Y ifF(x) ⊂ G(x) for every x ∈ X. The gauge of a multivalued mapping F :X → Y is defined as cal(F ) = sup{diamF(x) |x ∈ X}. The graph of multivalued mapping F :X → Y is the subset ΓF = {(x, y) ∈ X × Y | y ∈ F(x)} of the productX × Y . For arbitrary subset U ⊂ X × Y denote by U(x) the subset prY (U ∩ ({x} × Y)) of V . Then for the graph ΓF

we have ΓF (x) = F(x).A multivalued mapping G :X → Y is called complete if all sets {x} ×G(x) are closed with respect to some Gδ-set

S ⊂ X × Y containing the graph of this mapping. A multivalued mapping F :X → Y is called upper semicontinuousif for any open set U ⊂ Y the set {x ∈ X | F(x) ⊂ U} is open in X. A compact mapping is an upper semicontinuousmultivalued mapping with compact images of points.

An increasing sequence (finite or countably infinite) of subspaces

Z0 ⊂ Z1 ⊂ Z2 ⊂ · · · ⊂ Z

is called a filtration of space Z. A sequence of multivalued mappings {Fk :X → Y } is called a filtration of multivaluedmapping F :X → Y if for any x ∈ X, {Fk(x)} is a filtration of F(x).

We say that a filtration of multivalued mappings Gi :X → Y is complete (resp., compact) if every mapping Gi , iscomplete (resp., compact).

2. Local properties of multivalued mappings

Let γ be a property of a topological space such that every open subspace inherits this property: if a space X

satisfies γ , then any open subspace U ⊂ X also satisfies γ . We say that a space Z satisfies γ locally if every pointz ∈ Z has a neighborhood with this property.

N. Brodsky et al. / Topology and its Applications 155 (2008) 773–782 775

For a multivalued map F :X → Y to satisfy γ locally we not only require that every point-image F(x) has thisproperty locally, but for any points x ∈ X and y ∈ F(x) there exist a neighborhood W of y in Y and Y of x in X suchthat W ∩ F(x′) satisfies γ for every point x′ ∈ U . And we use the word “equi” for local properties of multivaluedmaps.

The first example of such a property is the local compactness.

Definition 2.1. A space X is called locally compact if every point x ∈ X has a compact neighborhood. We say that amultivalued map F :X → Y is equi locally compact if for any points x ∈ X and y ∈ F(x) there exists a neighborhoodW of y in Y and U of x in X such that W ∩ F(x′) is compact for every point x′ ∈ U .

Another local property we are going to use is the hereditary asphericity. Recall that a compactum K is calledapproximately aspherical if for any (equivalently, for some) embedding of K into an ANR-space Y every neighbor-hood U of K in Y contains a neighborhood V with the following property: any mapping of the sphere Sn into V ishomotopically trivial in U provided n � 2.

Definition 2.2. We call a space Z hereditarily aspherical if any compactum K ⊂ Z is approximately aspherical.A space Z is said to be locally hereditarily aspherical if any point z ∈ Z has a hereditarily aspheric neighborhood.

It is easy to prove that the 2-dimensional Euclidean space is hereditary aspherical. Note that any 2-dimensionalmanifold is locally hereditarily aspherical.

Definition 2.3. We say that a multivalued map F :X → Y is equi locally hereditarily aspherical if for any pointsx ∈ X and y ∈ F(x) there exists a neighborhood W of y in Y and U of x in X such that W ∩ F(x′) is hereditarilyaspherical for every point x′ ∈ U .

Now we consider different properties of pairs of spaces and define the corresponding local properties for spacesand multivalued maps. We follow definitions and notations from [8].

Definition 2.4. An ordering α of subsets of a space Y is proper provided:

(a) If W α V , then W ⊂ V ;(b) If W ⊂ V , and V α R, then W α R;(c) If W α V , and V ⊂ R, then W α R.

Definition 2.5. Let α be a proper ordering.

(a) A space Y is locally of type α if, whenever y ∈ Y and V is a neighborhood of y, then there a neighborhood W ofy such that W α V .

(b) A multivalued mapping F :X → Y is lower α-continuous if for any points x ∈ X and y ∈ F(x) and for anyneighborhood V of y in Y there exist neighborhoods W of y in Y and U of x in X such that (W ∩ F(x′)) α (V ∩F(x′)) provided x′ ∈ U .

For example, if W αV means that W is contractible in V , then locally of type α means locally contractible. Anothertopological property which arises in this manner is LCn (where W α V means that every continuous mapping of then-sphere into W is homotopic to a constant mapping in V ) and the corresponding lower α-continuity of multivaluedmap is called lower (n + 1)-continuity. For the special case n = −1 the property W α V means that V is nonempty,and lower α-continuity is the lower semicontinuity.

The following result is weaker than Lemma 3.5 from [4]. We will use it with different properties α in Section 4.

Lemma 2.6. Let a lower α-continuous mapping Φ :X → Y of compactum X to a metric space Y contains a compactsubmapping F . Then for any ε > 0 there exists a positive number δ such that for every point (x, y) ∈ O(ΓF , δ) wehave (O(y, δ) ∩ Φ(x)) α (O(y, ε) ∩ Φ(x)).


In order to use results from [17] we need a local property called polyhedral n-connectedness. A pair of spacesV ⊂ U is called polyhedrally n-connected if for any finite n-dimensional polyhedron M and its closed subpolyhedronA any mapping of A into V can be extended to a map of M into U . Note that for spaces being locally polyhedrallyn-connected is equivalent to be LCn−1 (it follows from Lemma 2.7). The corresponding local property of multivaluedmaps is called polyhedral lower n-continuity.

Lemma 2.7. Any lower n-continuous multivalued mapping is lower polyhedrally n-continuous.

Proof. The proof easily follows from the fact that in a connected filtration Z0 ⊂ Z1 ⊂ · · · ⊂ Zn of spaces the pair Z0 ⊂Zn is polyhedrally n-connected. Given a mapping f :A → Z0 of subpolyhedron A of n-dimensional polyhedron P ,we extend it successively over skeleta P (k) of P such that the image of k-dimensional skeleton P (k) is containedin Zk . Resulting map gives us an extension f :P → Zn of f which proves that the pair Z0 ⊂ Zn is polyhedrallyn-connected. �

A filtration of multivalued maps {Fi} is called polyhedrally connected if every pair Fi−1(x) ⊂ Fi(x) is polyhedrallyi-connected. A filtration {Fi} is called lower continuous if for any i the mapping Fi is lower i-continuous.

Lemma 2.8. If p :E → B is a Serre fibration of LC0-compacta with fibers homeomorphic to a given 2-dimensionalcompact manifold, then the multivalued mapping p−1 :B → E is

• equi locally hereditarily aspherical,• polyhedrally lower 2-continuous.

Proof. Since every open proper subset of a two-dimensional manifold is aspherical, every compact proper subset of2-manifold is approximately aspherical. Therefore, the mapping F is equi locally hereditarily aspherical.

It follows from a theorem of McAuley [13] that the mapping p−1 is lower 2-continuous. By Lemma 2.7, themapping p−1 is polyhedrally lower 2-continuous. �3. Singlevalued approximations

Definition 3.1. A singular pair of spaces is a triple (Z,φ,Z′) where φ :Z → Z′ is a mapping.We say that a space Z contains a singular filtration of spaces if a finite sequence of pairs {(Zi,φi)}ni=0 given where

Zi is a space and φi :Zi → Zi+1 is a map (we identify Zn+1 with Z).

For a multivalued map F :X → Y it is useful to consider its graph fibers {x} × F(x) ⊂ ΓF instead of usual fibersF(x) ⊂ Y . While the graph fibers are always homeomorphic to the usual fibers, different graph fibers do not intersect(the usual fibers may intersect in Y ). We denote the graph fiber of the map F over a point x ∈ X by FΓ (x).

To define the notion of singular filtration for multivalued maps we introduce a notion of fiberwise transformationof multivalued maps.

Definition 3.2. For multivalued mappings F and G of a space X a fiberwise transformation from F to G is a contin-uous mapping T :ΓF → ΓG such that T (FΓ (x)) ⊂ GΓ (x) for every x ∈ X.

A fiber T (x) of the fiberwise transformation T over the point x ∈ X is a mapping T (x) :F(x) → G(x) determinedby T .

We say that a multivalued mapping F :X → Y contains a singular filtration of multivalued maps if a finite sequenceof pairs {(Fi, Ti)}ni=0 is given where Fi :X → Yi is a multivalued mapping and Ti is a fiberwise transformation from Fi

to Fi+1 (we identify Fn+1 with F ).

To construct continuous approximations of multivalued maps we need the notion of approximate asphericity.

Definition 3.3. A pair of compacta K ⊂ K ′ is called approximately n-aspherical if for any embedding of K ′ intoANR-space Z for every neighborhood U of K ′ in Z there exists a neighborhood V of K such that any mappingf :Sn → V is homotopically trivial in U .

A compactum K is approximately n-aspherical if the pair K ⊂ K is approximately n-aspherical.


The following is a singular version of approximate asphericity.

Definition 3.4. A singular pair of compacta (K,φ,K ′) is called approximately n-aspherical if for any embeddingsK ⊂ Z and K ′ ⊂ Z′ in ANR-spaces and for any extension of φ to a map φ : OK → Z′ of some neighborhood OK ofK the following holds: for every neighborhood U of K ′ in Z′ there exists a neighborhood V of K in OK such that forany mapping f :Sn → V the spheroid φ ◦ f :Sn → U is homotopically trivial in U .

Following R.C. Lacher [12], one can prove that this notion does not depend neither on the choices of ANR-spacesZ and Z′ nor on the embeddings of K and K ′ into these spaces.

Definition 3.5. A singular filtration of compacta {(Ki,φi, )}ni=0 is called approximately connected if for every i < n

the singular pair (Ki,φi,Ki+1) is approximately i-aspherical.

Clearly, a singular pair of compacta (K,φ,K ′) is approximately n-aspherical in either of the following threesituations: compactum K , compactum K ′, or the pair φ(K) ⊂ K ′ is approximately n-aspherical.

Definition 3.6. A singular filtration F = {(Fi, Ti)}ni=0 of compact mappings Fi :X → Yi is said to be approximatelyconnected if for every point x ∈ X the singular filtration of compacta {(Fi(x), Ti(x))}ni=0 is approximately connected.

An approximately connected singular filtration F = {(Fi :X → Yi, Ti)}ni=0 is said to be approximately ∞-con-nected if the mapping Fn has approximately k-aspherical point-images Fn(x) for all k � n and all x ∈ X.

Note that if a singular filtration F = {(Fi, Ti)}ni=0 is approximately ∞-connected, then the mapping Fn containsan approximately connected singular filtration of any given finite length.

We will reduce our study of singular filtration s to the study of usual filtration s using the following cylinderconstruction.

Definition 3.7. For a continuous singlevalued mapping f :X → Y we define a cylinder of f denoted by cyl(f ) as aspace obtained from the disjoint union of X × [0,1] and Y by identifying each {x} × {1} with f (x).

Note that the cylinder cyl(f ) contains a homeomorphic copy of Y called the bottom of the cylinder, and a homeo-morphic copy of X as X × {0} called the top of the cylinder.

Remark 3.8. There is a natural deformation retraction r : cyl(f ) → Y onto the bottom Y . Clearly, the fiber of themapping r over a point y ∈ Y is either one point {y} or a cone over the set f −1(y). Therefore, if the map f is proper,then r is UV∞-mapping.

Remark 3.9. Suppose that X is embedded into Banach space B1 and Y is embedded into Banach space B2. Then wecan naturally embed the cylinder cyl(f ) into the product B1 × R × B2. The embedding is clearly defined on the topas embedding into B1 × {0} × {0} and on the bottom as embedding into {0} × {1} × B2. We extend these embeddingsto the whole cylinder by sending its point {x} × {t} to the point {(1 − t) · x} × t × {t · f (x)}.Lemma 3.10. If a singular pair of compacta (K,φ,K ′) is approximately n-aspherical, then the pair K ⊂ cyl(φ) isapproximately n-aspherical.

Proof. Let us fix embeddings of K into Banach space B1, of K ′ into Banach space B2, and of the cylinder cyl(φ) intothe product B = B1 × R × B2 as described in Remark 3.9. Fix a neighborhood U of cyl(φ) in B . Extend the mappingφ to a map φ1 :B1 → B2. Take a neighborhood V1 of the top of our cylinder in B1 such that the cylinder cyl(φ1|V1) iscontained in U . Using approximate n-asphericity of the pair (K,φ,K ′) we find for a neighborhood U ∩{0}×{1}×B2of K ′ in {0} × {1} × B2 a neighborhood V ′ of K in B1 × {0} × {0}. Let ε be a positive number such that the productV = V ′ × (−ε, ε) × O(0, ε) is contained in U .

Given a spheroid f :Sn → V we retract it into V ′ ×{0}×{0}, then retract it to the bottom of the cylinder cyl(φ1|V1)

using Remark 3.8, and finally contract it to a point inside U ∩{0}×{1}×B2. Clearly, the whole retraction sits inside U ,as required. �


Definition 3.11. Let F = {(Fi :X → Yi, Ti)}ni=0 be a singular filtration of a multivalued mapping F :X → Y = Yn+1.If all the spaces Yi are Banach, then for a multivalued mapping F from X to Y = Y ×∏n

i=0(Yi ×R) defined as F(x) =⋃nk=0 cyl(Tk(x)) we can define a cylinder cyl(F) as a filtration of multivalued maps {Fi}ni=0 defined as follows:

F0 = F0 and F(x) =i−1⋃

k=0

cyl(Tk(x)

).

It is easy to see that for a singular filtration F = {(Fi, Ti)}ni=0 of compact mappings Fi the filtration cyl(F) consistsof compact mappings Fi .

Lemma 3.12. If a singular filtration F = {(Fi, Ti)}ni=0 of compact maps is approximately connected, then the filtrationcyl(F) = {Fi}ni=0 is approximately connected.

Proof. Using Remark 3.8 it is easy to define a deformation retraction r : Fi (x) → Fi(x) which is UV∞-mapping. Thisretraction defines UV∞-mapping of pairs (Fi (x),Fi+1(x)) → (Fi(x), cyl(Ti(x))). By Lemma 3.10 the pair Fi(x) ⊂cyl(Ti(x)) is approximately i-aspherical, and by Pairs Mapping Lemma from [17] the pair Fi (x) ⊂ Fi+1(x) is alsoapproximately i-aspherical. �Theorem 3.13. Let H :X → Y be a multivalued mapping of metric space X to a Banach space Y . If dimX � n andH contains approximately connected singular filtration H = {(Hi :X → Yi, Ti)}ni=0 of compact mappings, then anyneighborhood U of the graph ΓH contains the graph of a single-valued and continuous mapping h :X → Y .

Proof. Without loss of generality, we may assume that all spaces Yi are Banach spaces. We consider Y as a subspaceof the product Y = Y × ∏n

i=0(Yi × R). Clearly, H is a submapping of a multivalued mapping H :X → Y definedas H(x) = ⋃n

k=0 cyl(Tk(x)) and ΓH admits a deformation retraction R onto ΓH . Fix a neighborhood U of the graphΓH in X × Y . Since all maps Hi are compact, H is also compact and the graph ΓH is closed in X × Y. Extend themapping prY ◦R :ΓH → Y to some neighborhood W of ΓH in X × Y and denote by R′ the map of W to X × Y suchthat prY ◦R′ is our extension. Clearly, we may assume that R′(W) is contained in U .

By Lemma 3.12 the multivalued map H admits approximately connected filtration cyl(H) of compact multivaluedmaps. By Single-Valued Approximation Theorem from [17] there exists a singlevalued continuous mapping h :X → Y

with Γh ⊂ W . Define a singlevalued continuous map h by the equality Γh = R′(Γh). Clearly, Γh, is contained inR′(W) ⊂ U . �Theorem 3.14. Suppose that a compact mapping of separable metric ANRs F :X → Y admits a compact singularapproximately ∞-connected filtration. Then for any compact space K ⊂ X every neighborhood of the graph ΓF (K)

contains the graph of a single-valued and continuous mapping f :K → Y .

Proof. Let U be an open neighborhood of the graph ΓF (K) in the product X × Y . Since F is upper semicontinuous,there is a neighborhood OK of compactum K such that ΓF (OK) is contained in U . Since any open subset of separableANR-space is separable ANR-space [11], we can denote OK by X and consider U as an open neighborhood of thegraph ΓF .

For every point x ∈ X take open neighborhoods Ox ⊂ X of the point x and Vx ⊂ X of the compactum F(x)

such that the product Ox × Vx is contained in U . Using upper semicontinuity of F we can choose Ox so small thatthe following inclusion holds: F(Ox) ⊂ Vx . Fix an open covering ω1 of the space X which is starlike refinement of{Ox}x∈X . Let ω2 be a locally finite open covering of the space X which is starlike refined into ω1.

There exist a locally finite simplicial complex L and mappings r :X → L and j :L → X such that the map j ◦ r

is ω2-close to idX [11]. Fix a finite subcomplex N ⊂ L containing the compact set r(K). Define a compact mappingΨ :N → Y by the formula Ψ = F ◦ j . Clearly, the mapping Ψ admits a compact approximately connected singularfiltration of any length (particularly, of the length dimN ). Let us define a neighborhood W of the graph ΓΨ . For everypoint q ∈ N we put

W(q) =⋂{

U(y) | y ∈ stω1

(Stω2

(j (q)

))}.


By Stω2(j (q)), we denote the star of the point j (q) with respect to the covering ω2. And by st(A,ω), we denote theset

⋃{U ∈ ω | A ⊂ U}.By Theorem 3.13 there exists a single-valued continuous mapping ψ :N → Y such that the graph Γψ is contained

in W . Put f = ψ ◦ r :X → Y . For any point x ∈ K we have ψ(r(x)) ∈ ⋂{U(x′) | x′ ∈ Stω2(j ◦ r(x))}. Since x ∈Stω2(j ◦ r(x)), then ψ(r(x)) ∈ U(x). That is, the graph of f is contained in U . �4. Fibrations with 2-manifold fibers

The following lemma is a weak form of Compact Filtration Lemma from [17].

Lemma 4.1. Any polyhedrally connected lower continuous finite filtration of complete mappings of a compact spacecontains a compact approximately connected subfiltration of the same length.

Lemma 4.2. Let F :X → Y be equi locally hereditarily aspherical, lower 2-continuous complete multivalued map-ping of ANR-space X to Banach space Y . Suppose that a compact submapping Ψ :A → Y of F |A is defined on acompactum A ⊂ X and admits continuous approximations. Then for any ε > 0 there exists a neighborhood OA ofA and a compact submapping Ψ ′ : OA → Y of FOA such that ΓΨ ′ ⊂ O(ΓΨ , ε), Ψ ′ admits a compact approximately∞-connected filtration, and calΨ ′ < ε.

Proof. Fix a positive number ε. Apply Lemma 2.6 with α being equi local hereditary asphericity to get a positive num-ber ε2 < ε/4. By Lemma 2.7 the mapping F is lower polyhedrally 2-continuous. Subsequently applying Lemma 2.6with α being polyhedral n-continuity for n = 2,1,0, we find positive numbers ε1, ε0, and δ such that δ < ε0 < ε1 < ε2and for every point (x, y) ∈ O(ΓΨ , δ) the pair (O(y, ε1) ∩ F(x),O(y, ε2) ∩ F(x)) is polyhedrally 2-connected, thepair (O(y, ε0) ∩ F(x),O(y, ε1) ∩ F(x)) is polyhedrally 1-connected, and the intersection O(y, ε0) ∩ F(x) is notempty.

Let f :K → Y be a continuous single-valued mapping whose graph is contained in O(ΓΨ , δ). Let f ′ :OK → Y

be a continuous extension of the mapping f over some neighborhood OK such that the graph of f ′ is contained inO(ΓΨ , δ). Now we can define a polyhedrally connected filtration G0 ⊂ G1 ⊂ G2 :OK → Y of the mapping F |OK

by the equality

Gi(x) = O(f ′(x), εi

) ∩ F(x).

Since the set⋃

x∈OK{{x} × O(f ′(x), εi)} is open in the product OK × Y and the mapping F is complete, then Gi

is also complete. Clearly, calG2 < 2ε2 < ε and for any point x ∈ K the set GΓ2 (x) is contained in O(ΓΨ , ε). Now,

applying Lemma 4.1 to the filtration G0 ⊂ G1 ⊂ G2, we obtain a compact approximately connected subfiltrationF0 ⊂ F1 ⊂ F2 :OK → Y . By the choice of ε2 the mapping F2 has approximately aspherical point-images. Therefore,the filtration F0 ⊂ F1 ⊂ F2 is approximately ∞-connected. Finally, we put Ψ ′ = F2. �Theorem 4.3. Let F :X → Y be equi locally hereditarily aspherical, lower 2-continuous complete multivalued map-ping of locally compact ANR-space X to Banach space Y . Suppose that a compact submap-ping Ψ :A → Y of F |A isdefined on compactum A ⊂ X and admits continuous approximations. Then for any ε > 0 there exists a neighborhoodOA of A and a single-valued continuous selection s : OA → Y of F |OA such that Γs ⊂ O(ΓΨ , ε).

Proof. Consider a Gδ-subset G ⊂ X × Y such that all fibers of F are closed in G and fix open sets Gi ⊂ X × Y

such that G = ⋂∞i=1 Gi . Fix ε > 0 such that O(ΓΨ , ε) ⊂ G1. By Lemma 4.2 there is a neighborhood U1 of A in X

and a compact submapping Ψ1 :U1 → Y of F |U1 such that ΓΨ1 ⊂ O(ΓΨ , ε), Ψ1 admits a compact approximately ∞-connected filtration, and calΨ1 < ε. Since X is locally compact and A is compact, there exists a compact neighborhoodOA of A such that OA ∈ U1. By Theorem 3.14 the mapping Ψ1|OA admits continuous approximations. Take ε1 < ε

such that the neighborhood U1 = O(ΓΨ1(OA), ε1) lies in O(ΓΨ , ε). Clearly, U1 ⊂ G1.Now by induction with the use of Lemma 4.2, we construct a sequence of neighborhoods U1 ⊃ U2 ⊃ U3 ⊃ · · ·

of the compactum OA, a sequence of compact submappings {Ψk :Uk → Y }∞k=1 of the mapping F , and a sequenceof neighborhoods Uk = O(ΓΨ1(OA), ε) such that for every k � 2 we have calΨk < εk−1/2 < ε/2k , and Uk(OA) is


contained in Uk−1(OA) ∩ Gk . It is not difficult to choose the neighborhood Uk of the graph ΓΨkin such a way that for

every point x ∈ Uk the set Uk(x) has diameter less than 32k .

Then for any m � k � 1 and for any point x ∈ OA we have Ψm(x) ⊂ O(Ψk(x), 32k ); this implies the fact that the se-

quence {Ψk|OA}∞k=1 is a Cauchy sequence. Since Y is complete, there exists the limit s : OA → Y of this sequence. Themapping s is single-valued by the condition calΨk < 1

2k and is upper semicontinuous (and, therefore, is continuous)by the upper semicontinuity of all the mappings Ψk . Clearly, for any x ∈ OA the point s(x) lies in G(x) and is a limitpoint of the set F(x). Since F(x) is closed in G(x), then s(x) ∈ F(x), i.e. s is a selection of the mapping F . �Theorem 4.4. Let p :E → B be a Serre fibration of locally connected compacta with all fibers homeomorphic to somefixed two-dimensional manifold. If B ∈ ANR, then any section of p over closed subset A ⊂ B can be extended to asection of p over some neighborhood of A.

Proof. Let s :A → E be a section of p over A. Embed E into Hilbert space l2 and consider a multivalued mappingF :B → l2 defined as follows:

F(b) ={

s(b), if b ∈ A,

p−1(b), if x ∈ B \ A.

Since every fiber p−1(b) is compact, the mapping F is complete. By Lemma 2.8 the mapping F is equi locallyhereditarily aspherical and lower 2-continuous. We can apply Theorem 4.3 to the mapping F and its submapping s tofind a single-valued continuous selection s : OA → l2 of F |OA. By definition of F , we have s|A = F |A = s. Clearly, s

defines a section of the fibration p over OA extending s. �Definition 4.5. For a mapping p :E → B we say that s :B → E is ε-section if the map p ◦ s is ε-close to theidentity idB .

The following proposition easily follows from Theorem 4.1 of the paper [14].

Proposition 4.6. If p :E → B is a locally trivial fibration of finite-dimensional compacta with locally contractiblefiber, then there is ε > 0 such that an existence of ε-section for p implies an existence of a section for p.

We will use the following two propositions in the proof of existence of global sections in Serre fibrations. Forthe definition and basic properties of Menger manifolds we refer the reader to [2]. Proofs of these two propositionsfollow from Bestvina’s construction of Menger manifold [2] and Dranishnikov’s triangulation theorem for Mengermanifolds [6].

Proposition 4.7. Let X be a compact 2-dimensional Menger manifold. For any ε > 0 there exist a finite polyhedronP and maps g :X → P and h :P → X such that h ◦ g is ε-close to the identity. If π1(X) = 0, then we may choose P

with π1(P ) = 0.

Proposition 4.8. Let X be a compact 3-dimensional Menger manifold with π1(X) = π2(X) = 0. For any ε > 0 thereexist a finite polyhedron P with π1(P ) = π2(P ) = 0 and maps g :X → P and h :P → X such that h ◦ g is ε-close tothe identity.

Theorem 4.9. Let p :E → B be a Serre fibration of locally connected compactum E onto an ANR-compactum B

with all fibers homeomorphic to a given two-dimensional manifold M . If M is not homeomorphic to the sphere or theprojective plane, then p admits a global section provided one of the following conditions holds:

(a) π1(M) is abelian and H 2(B;π1(Fb)) = 0.(b) π1(M) is non-abelian, M is not homeomorphic to the Klein bottle and π1(B) = 0.(c) M is homeomorphic to the Klein bottle and π1(B) = π2(B) = 0.


Proof. Embed E into the Hilbert space l2 and consider a multivalued mapping F :B → l2 defined as F = p−1. Sinceevery fiber p−1(b) is compact, the mapping F is complete. It follows from Lemma 2.8 that the mapping F is equilocally hereditarily aspherical and lower 2-continuous.

Now we show that F admits a compact singular approximately ∞-connected filtration. In cases (a) and (b) thereexists UV1-mapping μ of Menger 2-dimensional manifold L onto B [5]. Note that π1(L) = 0 if π1(B) = 0. In case(c) we consider UV2-mapping μ of Menger 3-dimensional manifold L onto B [5]; note that π1(L) = π2(L) = 0if π1(B) = π2(B) = 0. Since dimL < ∞, the induced fibration pL = μ∗(p) :EL → L is locally trivial [10]. ByProposition 4.6 there is ε > 0 such that an existence of ε-section for pL implies an existence of a section for pL. Incases (a) and (b), by Proposition 4.7, there exist a 2-dimensional finite polyhedron P and continuous maps g :L → P

and h :P → L such that h ◦ g is ε-close to the identity (we assume π1(P ) = 0 in case π1(B) = 0). In case (c)by Proposition 4.8 there exist a 3-dimensional finite polyhedron P with π1(P ) = π2(P ) = 0 and continuous mapsg :L → P and h :P → L such that h ◦ g is ε-close to the identity.

Consider a locally trivial fibration pP = h∗(pL) :EP → P .

Claim. The fibration pP has a section sP .

Proof. (a) If π1(M) is abelian and H 2(B;π1(Fb)) = 0 = H 2(P ;π1(Fb)), the fibration pP has a section sP [18].(b) Since π1(P ) = 0 and dimP = 2, then P is homotopy equivalent to a bouquet of 2-spheres Ω = ∨m

i=1 S2i . Let

ψ :P → Ω and φ :Ω → P be maps such that φ ◦ ψ is homotopic to the identity idP . The locally trivial fibrationover a bouquet pΩ = φ∗(pP ) :EΩ → Ω has a global section if and only if it has a section over every sphere ofthe bouquet. If the fiber M has non-abelian fundamental group and is not homeomorphic to Klein bottle, then thespace of autohomeomorphisms Homeo(M) has simply connected identity component [1] and therefore any locallytrivial fibration over 2-sphere with fiber homeomorphic to M has a section (in fact, this fibration is trivial). Hence, thefibration pΩ has a section sΩ . This section defines a lifting of the map φ ◦ ψ :P → P with respect to pP . Since pP

is a Serre fibration and φ ◦ ψ is homotopic to the identity, the identity mapping idP has a lifting sP :P → EP withrespect to pP which is simply a section of pP .

(c) Since π1(P ) = π2(P ) = 0 and dimP = 3, then P is homotopy equivalent to a bouquet of 3-spheres Ω =∨mi=1 S3

i . Let ψ :P → Ω and φ :Ω → P be maps such that φ ◦ ψ is homotopic to the identity idP . The locally trivialfibration over the bouquet pΩ = φ∗(pP ) :EΩ → Ω has a global section if and only if it has a section over every sphereof the bouquet. Since the space of autohomeomorphisms of the Klein bottle Homeo(K2) has π2(Homeo(K2)) = 0[1], any locally trivial fibration over 3-sphere with fiber homeomorphic to K2 has a section (in fact, this fibration istrivial). Hence, the fibration pΩ has a section sΩ . This section defines a lifting of the map φ ◦ψ :P → P with respectto pP . Since pP is a Serre fibration and φ ◦ ψ is homotopic to the identity, the identity mapping idP has a liftingsP :P → EP with respect to pP which is simply a section of pP . �

By the construction of P the section sP defines an ε-section for pL. Therefore, pL has a section sL. Clearly, sLdefines a lifting T :L → E of μ with respect to p. Finally, we define compact singular filtration F = {(Fi, Ti)}2

i=0 ofF as follows:

F0 = F1 = μ−1 :B → L, F2 = F, Ti = id for i = 0

and T1 is defined fiberwise by T1(x) = T |μ−1(x) :μ−1(x) → F(x). The filtration F is approximately connected sincefor i = 0,1 any compactum Fi(x) is UV1. And F is approximately ∞-connected since every compactum F(x) is anaspherical 2-manifold (and therefore is approximately n-aspherical for all n � 2).

Now we can apply Theorem 4.3 to the mapping F to find a single-valued continuous selection s :B → l2 of F .Clearly, s defines a section of the fibration p. �

The following remark explains the appearance of the condition (c) in Theorem 4.9.

Remark 4.10. There exists a locally trivial fibration over 2-sphere with fibers homeomorphic to Klein bottle havingno global section.


Acknowledgements

Authors wish to express their sincere thanks to P. Akhmetiev, R.J. Daverman, B. Hajduk and T. Yagasaki for helpfuldiscussions during the development of this work.

References

[1] W. Balcerak, B. Hajduk, Homotopy type of automorphism groups of manifolds, Colloq. Math. 45 (1981) 1–33.[2] M. Bestvina, Characterizing k-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (380) (1988).[3] N. Brodsky, Sections of maps with fibers homeomorphic to a two-dimensional manifold, Topology Appl. 120 (2002) 77–83.[4] N. Brodsky, A. Chigogidze, A. Karasev, Approximations and selections of multivalued mappings of finite-dimensional spaces, JP J. Geom.

Topol. 2 (2002) 29–73.[5] A.N. Dranishnikov, Absolute extensors in dimension n and n-soft mappings increasing the dimension, Russian Math. Surveys 39 (5) (1984)

63–111.[6] A.N. Dranishnikov, Universal Menger compacta and universal mappings, Math. USSR Sb. 57 (1) (1987) 131–150 (in Russian).[7] A.N. Dranishnikov, E.V. Šcepin, Cell-like mappings. The problem of increase of dimension, Russian Math. Surveys 41 (6) (1986) 59–111.[8] J. Dugundji, E. Michael, On local and uniformly local topological properties, Proc. Amer. Math. Soc. 7 (1956) 304–307.[9] L. Gorniewicz, A. Granas, W. Kryszewski, On the homotopy method in the fixed point index theory of multi-valued mappings of compact

absolute neighborhood retracts, J. Math. Anal. Appl. 161 (1991) 457–473.[10] M.E. Hamstrom, E. Dyer, Regular mappings and the space of homeomorphisms on a 2-manifold, Duke Math. J. 25 (1958) 521–531.[11] O. Hanner, Some theorems on absolute neighborhood retracts, Arch. Math. 1 (1951) 389–408.[12] R.C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977) 495–552.[13] F. McAuley, P.A. Tulley, Fiber spaces and n-regularity, in: Topology Seminar Wisconsin, in: Ann. of Math. Studies, vol. 60, Princeton Univ.

Press, 1965.[14] E. Michael, Continuous selections, II, Ann. Math. 64 (1956) 562–580.[15] D. Repovš, P.V. Semenov, E.V. Šcepin, Topologically regular maps with fibers homeomorphic to a one-dimensional polyhedron, Houston J.

Math. 23 (1997) 215–230.[16] E.V. Šcepin, On homotopically regular mappings of manifolds, in: Banach Centre Publ., vol. 18, PWN, Warszawa, 1986, pp. 139–151.[17] E.V. Šcepin, N. Brodsky, Selections of filtered multivalued mappings, Proc. Steklov Inst. Math. 212 (1996) 218–228.[18] G.W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York–Berlin, 1978.[19] H. Whitney, Regular families of curves, Ann. Math. (2) 34 (1933) 244–270.



L’espace des sélections d’une dendrite

Robert Cauty

Université Paris 6, Institut de mathématiques de Jussieu, case 247, 4 place Jussieu, 75252 Paris Cedex 05, France

Reçu le 17 novembre 2006 ; reçu en forme révisée le 30 novembre 2006 ; accepté le 30 novembre 2006

Abstract

We prove that the selection space of a nondegenerate dendrite is homeomorphic to �2.© 2007 Elsevier B.V. All rights reserved.

MSC : 54C65 ; 54F50 ; 54F65

Keywords: Selection; Dendrite; Hilbert space

1. Introduction

Tous les espaces considérés dans cet article sont supposés métrisables.Pour tout continu X, nous notons C(X) l’espace des sous-continus non vides de X muni de la topologie de Vietoris.

Une fonction σ : C(X) → X est une sélection si elle est continue et vérifie σ(K) ∈ K pour tout K ∈ C(X). Nousnotons Σ(X) l’espace de toutes les sélections de X, muni de la topologie de la convergence uniforme. Cet espacea été introduit dans [3] par J.E. McParland. Nous nous proposons ici de prouver le théorème suivant, qui résout leproblème 44 de [2].

Théorème 1. Si D est une dendrite non dégénérée, alors Σ(D) est homéomorphe à �2.

Nous notons I l’intervalle [0,1], In le produit de n copies de I et⊕

In la somme topologique des In, n � 1. Unefamille {Fn | n � 1} de sous-ensembles d’un espace X est dite discrète si tout point de X a un voisinage rencontrantau plus l’un des Fn.

Nous utiliserons la caractérisation suivante de �2, due à Torunczyk [4] (voir aussi [1]), dans laquelle d est unedistance arbitraire, mais fixée, définissant la topologie de X.

Théorème 2. Un espace X est homéomorphe à �2 si, et seulement si, c’est un rétracte absolu séparable topologique-ment complet vérifiant la condition suivante :

(�) Pour toute fonction continue f : ⊕In → X et toute fonction continue ε : X →]0,1], il existe une fonction

continue g : ⊕ In → X telle que d(f (x), g(x)) < ε(f (x)) pour tout x ∈ ⊕In, et que la famille {g(In) | n � 1}

soit discrète.

Adresse e-mail : [email protected].


784 R. Cauty / Topology and its Applications 155 (2008) 783–786

Puisque D est un rétracte absolu compact et C(D) compact, l’espace DC(D) des fonctions continues de C(D)

dans D, muni de la topologie de la convergence uniforme, est un rétracte absolu séparable topologiquement complet.Il est remarqué dans [3] (Théorème 3.3) que Σ(D) est un rétracte de DC(D), donc Σ(D) est un rétracte absoluséparable topologiquement complet, et il ne rest plus qu’à montrer qu’il possède la propriété (�), ce que nous feronsà la Section 3, après quelque préliminaires.

2. Préliminaires

Soit D une dendrite non dégénérée. Si p, q sont deux points de D, nous notons [p,q] l’unique arc d’extrémitésp,q contenu dans D, en convenant que [p,q] = {p} si p = q . Puisque D est localement connexe, sa topologie peutêtre définie par une distance convexe. Fixons une telle distance d , que nous supposerons majorée par un. Alors, quelsque soient p et q dans D, [p,q] est isométrique à [0, d(p, q)]. Pour s ∈ I , soit λ(p,q, s) le point de l’arc [p,q] telque d(p,λ(p,q, s)) = sd(p, q). La fonction λ vérifie λ(p,q,0) = p, λ(p,q,1) = q et λ(p,p, s) = p pour tout s. Lelemme suivant est connu (voir la démonstration du Lemme 2.5 de [3]).

Lemme 1. La fonction λ : D × D × I → D est continue.

Lemme 2. Soient Y un espace métrisable, F un fermé de C(D) × Y et α : F → D une fonction continue telle queα(K,y) ∈ K pour tout (K,y) ∈ F . Il existe une fonction continue β : C(D) × Y → D qui prolonge α et vérifieβ(K,y) ∈ K pour tout (K,y) ∈ C(D) × Y .

Démonstration. Pour K ∈ C(D) et p ∈ D, soit [p,R(K,p)] l’arc irréductible entre p et K (dégénéré si p ∈ K). Lafonction R : C(D) × D → D est continue ([3], Lemme 3.2). Puisque D est un rétracte absolu, α peut se prolonger enune fonction continue γ : C(D) × Y → D. Alors la fonction β : C(D) × Y → D définie par

β(K,y) = R(K,γ (K,y)

)

est continue et a les propriétés souhaitées. �Nous notons ρ la distance de la convergence uniforme sur DC(D), et δ(A) le diamètre d’un sous-ensemble non

vide A de D.

3. Démonstration du Théorème 1

Comme nous l’avons remarqué, il suffit de vérifier que Σ(D) possède la propriété (�). Soient donc f : ⊕In →

Σ(D) et ε : Σ(D) →]0,1] des fonctions continues. Alors la fonction f : C(D)× (⊕

In) → D définie par f (K,x) =f (x)(K) est continue.

Soient A0, A1 deux arcs disjoints contenus dans D, où Ai = [pi, qi] pour i = 0,1, et soit αi un homéomorphismede I sur Ai tel que d(pi,αi(s)) = sd(pi, qi) pour tout s ∈ I .

Pour n � 1 entier et 0 � u � v � 1, la formule

τn0

(α0

([u,v])) = α0

((1 − 1

n

)u + 1

nv

)

définit une sélection sur C(A0), et le lemme 2 nous permet d’étendre τn0 en une sélection σn

0 sur C(D). Remarquonsque

(1) Si K est un sous-continu non dégénéré de A0, alors σn0 (K) �= σm

0 (K) pour n �= m.

Pour n � 1 et 0 < s � 1, posons L(s) = α1([0, s/2]) et Mn(s) = α1([s/3n, s/2]). L’ensemble

Fn =⋃ {

L(s),Mn(s)} × {s}

0<s�1

R. Cauty / Topology and its Applications 155 (2008) 783–786 785

est fermé dans C(D)×]0,1]. Posant

τn1

(L(s), s

) = α1(s/2)

et

τn1

(Mn(s), s

) = α1(s/3n),

nous obtenons une fonction continue τn1 : Fn → D telle que τn

1 (K, s) ∈ K pour tout (K, s) ∈ Fn. Le Lemme 2 nouspermet de prolonger τn

1 en une fonction contiue σn1 : C(D)×]0,1] → D telle que σn

1 (K, s) ∈ K pour tout (K, s) ∈C(D)×]0,1].

Pour s, t ∈ I , nous avons d(α1(s), α1(t)) = |s − t |d(p1, q1) � |s − t |, d’où, pour tout s ∈]0,1] et tout n � 1,

(2) δ(L(s)) � s/2 et δ(Mn(s)) < s/2.

Les ensembles C(A0) et C(A1) étant disjoints et fermés dans C(D), nous pouvons trouver des ouverts disjointsO0 et O1 de C(D) tels que C(Ai) ⊂ Oi pour i = 0,1. Posons

Hi = {(K, s) ∈ C(Ai)×]0,1] | δ(K) � s/2

}

et

Ui = {(K, s) ∈ Oi×]0,1] | δ(K) < 3s/4

}.

L’ensemble Ui est ouvert dans C(D)×]0,1] et contient Hi , qui est fermé. Nous pouvons donc trouver un ouvertVi de C(D)×]0,1] tel que Hi ⊂ Vi et V i ⊂ Ui . Soit γi : C(D)×]0,1] → I une fonction continue nulle sur Hi etégale à un sur le complémentaire de Vi .

Pour n � 1, définissons une fonction gn : C(D) × In → D comme suit

gn(K,x) =⎧⎨

⎩

λ(σn0 (K), f (K,x), γ0(K, ε(f (x)))) si (K, ε(f (x))) ∈ U0,

λ(σn1 (K, ε(f (x))), f (K,x), γ1(K, ε(f (x)))) si (K, ε(f (x))) ∈ U1,

f (K,x) si (K, ε(f (x))) /∈ V 0 ∪ V 1.

Comme O0 et O1 sont disjoints, il en est de même de U0 et U1. Si (K, ε(f (x))) ∈ U0 \V 0, alors γ0(K, ε(f (x))) =1 et la première formule donne

gn(K,x) = λ(σn

0 (K), f (K,x),1) = f (K,x).

De même, les deux dernières formules coïncident si (K, ε(f (x))) ∈ U1 \ V 1. La fonction gn est donc bien définie.En outre, l’ensemble des (K,x) tels que (K, ε(f (x))) appartienne à U0 (resp., U1, resp., C(D)×]0,1] \ (V 0 ∪ V 1))est un ouvert sur lequel gn est continue ; comme ces trois ouverts recouvrent C(D) × In, gn est continue.

Si gn(K,x) �= f (K,x), il existe i ∈ {0,1} tel que (K, ε(f (x))) ∈ Ui . Si (K, ε(f (x))) ∈ U0, alors gn(K,x) appar-tient à l’arc [σn

0 (K), f (K,x)]. Comme σn0 (K) et f (K,x) appartiennent au continu K , cet arc est contenu dans K ,

donc gn(K,x) ∈ K et, d’après la définition de U0,

d(gn(K,x), f (K,x)

)� δ(K) <

3

4ε(f (x)

).

Si (K, ε(f (x))) ∈ U1, un argument analogue montre que gn(K,x) ∈ K et que d(gn(K,x), f (K,x)) < 34ε(f (x)),

donc d(gn(K,x), f (K,x)) < 34ε(f (x)) pour tout K ∈ C(D).

Comme gn est continue, la fonction gn : In → DC(D) définie par gn(x)(K) = gn(K,x) est continue. D’après cequi précède, pour tout x ∈ In, gn(x) ∈ Σ(D) et nous avons

ρ(gn(x), f (x)

) = sup{d(gn(K,x), f (K,x)

) | K ∈ C(D)}

� 3

4ε(f (x)

).

Puisque⊕

In est la somme topologique des In, nous définissons une fonction continue g : ⊕ In → Σ(D) en posantg|In = gn pour tout n. La condition ρ(f (x), g(x)) < ε(f (x)) est alors vérifiée, et il ne reste plus qu’à montrer que lafamille {g(In) | n � 1} est discrète.

786 R. Cauty / Topology and its Applications 155 (2008) 783–786

Soient x ∈ In et y ∈ Im tels que g(x) = g(y). Soit K un sous-continu non dégénéré de A0 tel que δ(K) <

min{ 12ε(f (x)), 1

2ε(f (y))}. Alors (K, ε(f (x))) apartient à H0, donc γ0(K, ε(f (x))) = 0, et nous avons

g(x)(K) = gn(K,x) = λ(σn

0 (K), f (K,x),0) = σn

0 (K),

et de même

g(y)(K) = σm0 (K).

Puisque g(x) = g(y), il résulte de (1) que n = m. Les ensembles g(In), n � 1, sont donc deux à deux disjoints.Puisque {g(In) | n � 1} est une famille d’ensembles disjoints, si elle n’est pas discrète dans Σ(D), il existe une suitestrictement croissante {nk}∞k=1 d’entiers et, pour tout k, un point xk ∈ Ink tels que {g(xk)} converge vers un élément h

de Σ(D).Soit εk = ε(f (xk)). Quitte à passer à une sous-suite, nous pouvons supposer que la suite {εk} converge vers ε0 ∈ I .

Si ε0 = 0, alors, puisque ρ(f (xk), g(xk)) < εk , la suite {f (xk)} converge aussi vers h, et la continuité de la fonctionε entraîne que {εk} converge vers ε(h) > 0, ce qui est contradictoire.

Supposons donc ε0 > 0. D’après (2), (L(εk), εk) et (Mnk(εk), εk) appartiennent à H1, donc γ1(L(εk), εk) = 0 =

γ1(Mnk(εk), εk), et nous avons

g(xk)(L(εk)

) = λ(σ

nk

1

(L(εk), εk

), f

(L(εk), xk

),0

) = α1(εk/2)

et

g(xk)(Mnk

(εk)) = λ

(σ

nk

1

(Mnk

(εk), εk

), f

(Mnk

(εk), xk

),0

)

= σnk

1

(Mnk

(εk), εk

) = α1(εk/3nk).

Quand k tend vers l’infini, {L(εk)} et {Mnk(εk)} tendent vers K0 = α1([0, ε0/2]) et, comme {g(xk)} converge

uniformément vers h, les suites {g(xk)(L(εk))} et {g(xk)(Mnk(εk))} tendent vers h(K0). Mais {α1(εk/2)} tend vers

α1(ε0/2) et {α1(εk/3nk)} tend vers α1(0). Comme α1(0) �= α1(ε0/2), ceci est absurde.

Références

[1] M. Bestvina, P. Bowers, J. Mogilski, J. Walsh, Characterization of Hilbert space manifolds revisited, Topology Appl. 24 (1986) 53–69.[2] V. Martínez-de-la-Vega, J.M. Martínez-Montejano, Open problems on dendroids, in: R. Pearl (Ed.), Open Problems in Topology II, 2007,

pp. 319–324.[3] J.E. McPerland, The selection space of a dendroid, I, in: Continuum Theory (Denton, TX, 1999), Marcel Dekker Inc., New Yok, 2002, pp. 245–

269.[4] H. Torunczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981) 247–262.



Reduction principles in the theory of selections

Mitrofan M. Choban

Department of Mathematics, Tiraspol State University, str. Gh. Iablocichin 5, Chisinau, MD-2069, Moldova

Received 18 October 2006; received in revised form 25 April 2007; accepted 25 April 2007

Dedicated to Professor Ernest Michael on the occasion of his 80th birthday

Abstract

In the present article many theorems about existence of continuous selections for a set-valued mapping into the metrizable spacesare extended to mappings into non-metrizable spaces.© 2007 Published by Elsevier B.V.

MSC: 54C60; 54C10; 54C65; 54B20; 46A55; 46C50; 47H04

Keywords: Metrizable family; Reduction principle; Linear space; Topological group; Set-valued mapping; Selection

0. Introduction

The theory of selections for set-valued mappings appeared in 1956 in the papers [10–14] of E. Michael. In 2006we are celebrating half a century of this theory. It is remarkable that the founder of the theory of selections, ProfessorErnest Michael, has recently celebrated his 80th birthday.

In fact, we should mention that in 1951 E. Michael [9] developed the theory of spaces of subsets which has openedthe door to the theory of set-valued mappings: to every topology on the spaces of subsets corresponds a concretenotion of continuity of the set-valued mappings.

In the theory of selections, as a rule, the range space is metrizable and the domain space is paracompact. Theobjective of the reduction principles is to extend these results under more general requirements.

The principle of equivalence of set-valued mappings permits to replace the condition of metrization of the rangespace by condition that the family of images of points is metrizable. The notion of a metrizable family of subsets of auniform space was introduced by V.A. Geiler [7]. In [1] we defined this notion for arbitrary topological spaces. Thefollowing examples are important and useful:

1. The family of singleton subsets of an arbitrary space is complete metrizable.2. If H is a metrizable subgroup of a topological group G, then the family {xH : x ∈ G} is metrizable.3. A single-valued mapping is equivalent to a mapping into a singleton space.

E-mail address: [email protected].

0166-8641/$ – see front matter © 2007 Published by Elsevier B.V.doi:10.1016/j.topol.2007.04.015

788 M.M. Choban / Topology and its Applications 155 (2008) 787–796

We shall use the terminology from [6,19]. The author thanks the referee for a careful reading and a lot of helpfuladvice.

1. Set-valued mappings

Let X and Y be two nonempty topological spaces. A set-valued or a multi-valued mapping θ :X → Y assigns toeach point x ∈ X a nonempty subset θ(x) of Y . If for every x ∈ X the set θ(x) is a singleton set, then θ :X → Y is asingle-valued mapping.

If ϕ,ψ :X → Y are set-valued mappings and ϕ(x) ⊆ ψ(x) for any x ∈ X, then ϕ is a selection of ψ .Let θ :X → Y be a set-valued mapping, A ⊆ X and B ⊆ Y . The set θ(A) = ⋃{θ(x): x ∈ A} is the image of the

set A and the set θ−1(B) = {x ∈ X: θ(x) ∩ B �= ∅} is the inverse image of the set B. If Y = θ(X), then the inversemapping θ−1 :Y → X of the mapping θ is defined.

A set-valued mapping θ :Y → X is called lower (upper) semi-continuous if for every open (closed) subset H ofY the set θ−1(H) is open (closed) in X. The mapping θ is continuous if θ is lower and upper semi-continuous. Themapping θ is open (closed) if for every open (closed) subset L of X the set θ(L) is open (closed) in Y . The subspaceGr(θ) = ⋃{{x} × θ(x): x ∈ X} of the space X × Y is the graph of the mapping θ . For every point (x, y) ∈ Gr(θ)

we put θX(x, y) = x and θY (x, y) = y. Then θX : Gr(θ) → X, θY : Gr(θ) → Y are continuous single-valued mappingsand θ(x) = θY (θ−1

X (x)) for every x ∈ X, i.e. θ = θY ◦ θ−1X .

The following two assertions are proved in [3].

Proposition 1.1. For the mapping θ :X → Y the following statements are equivalent:

1. θ :X → Y is lower semi-continuous.2. θX : Gr(θ) → X is open.

Proposition 1.2. For the mapping θ :X → Y the following statements are equivalent:

1. θ :X → Y is upper semi-continuous and for every point x ∈ X the set θ(x) is compact.2. θX : Gr(θ) → X is a perfect mapping.

If the mapping θX is closed, then the mapping θ is upper semi-continuous. There exists an upper semi-continuousmapping θ :X → Y of a countable compact space X into a countable discrete space Y for which the mapping θX isnonclosed (see [3]). In the following example we propose a general method of construction of mappings of this kind.

Example 1.3. Let X be a completely regular space, {Uα: α ∈ A} be a family of open pairwise disjoint subsets of X,bα ∈ Uα for any α ∈ A and the set B = {bα: α ∈ A} be nonclosed in X. Assume that A is equipped with the discretetopology. We put Y = A × [0,1]. For every α ∈ A fix a continuous function fα :X → [0,1] where fα(bα) = 1 andX \ Uα ⊆ f −1

α (0). Let F = ⋂{f −1α (0): α ∈ A} and Vα = Uα \ F for any α ∈ A. Now we put θ(x) = Y for any

x ∈ F and θ(x) = {(α,fα(x))} for all α ∈ A and x ∈ Vα . The mapping θ is upper semi-continuous and the projectionθX : Gr(θ) → X is nonclosed.

The space Y is metrizable and locally compact. We mention the following special cases:Case 1. The space X is countably compact or contains a non-trivial convergent sequence.In this case we can assume that the set A is countable. We may suppose that Y is a closed subspace of the space of

reals.Case 2. The sets Uα , α ∈ A, are open-and-closed in X or indX = 0.In this case we assume that Y = A × {0,1} is a discrete space, X \ Uα ⊆ X \ f −1

α (1) = f −1α (0). In this case the

mapping θ is open and closed.

2. Problem of selections

We say that a property P of set-valued mappings is “nice” if every set-valued ϕ :X → Y with the property Psatisfies the following two conditions:

M.M. Choban / Topology and its Applications 155 (2008) 787–796 789

N1. The set ϕ(x) is compact for every x ∈ X, i.e. ϕ is a compact-valued mapping.N2. The property P is preserved by the relation of the equivalence of set-valued mappings in the sense of Defini-

tion 6.1.

2.1. General problem

Under which conditions for a set-valued mapping θ :X → Y does there exist a selection with the given “nice”property P?

Several results on the existence of “nice” selections (see [1,2,5,7,8,10–25]) are obtained under the following con-ditions:

II. X is a Hausdorff paracompact space.LC. θ is a lower semi-continuous mapping.

CM. The space Y is metrizable by a metric ρ such that every subspace θ(x) is complete metrizable by the metric ρ.

3. Notion of reduction

Let A, B be two classes of spaces and Ω be a class of set-valued mappings.We say that for a set-valued mapping θ :X → Y there exists an (A,B,Ω)-reduction if there exist two spaces

X1 ∈ A, Y1 ∈ B and a mapping θ1 :X1 → Y1 ∈ Ω such that for the mapping θ there exists a concrete “nice” selectionif and only if a selection with the same property exists for the mapping θ1.

Our purpose is to suggest some reduction principles.

4. Principle of paracompact extension

A set-valued mapping θ :X → Y admits a paracompact extension if X is a completely regular space and there exista paracompact space Z and a lower semi-continuous mapping ψ :Z → Y such that X is a subspace of the space Z

and θ = ψ |X, i.e. θ(x) = ψ(x) for any x ∈ X ⊆ Z.A set-valued mapping θ admits a strong paracompact extension if there exist a paracompact space Z and a lower

semi-continuous mapping ψ :Z → Y such that X is a subspace of the space Z, θ = ψ |X and for every z ∈ Z thereexists a point x = x(z) ∈ X such that ψ(z) ⊆ θ(x).

Proposition 4.1. Let ψ :Z → Y be a lower semi-continuous extension of a mapping θ :X → Y and X be a subspaceof the completely regular space Z. Then there exist a space S and a lower semi-continuous mapping ϕ :S → Y suchthat:

1. X is a dense subspace of the space S and X ⊆ S ⊆ βX.2. There exists a perfect single-valued mapping g :S → Z1 into a closed subspace Z1 of the space Z such that

ϕ = ψ ◦ g.3. If Z is paracompact, then S is paracompact, too.4. ϕ is an extension of the mapping θ .5. If ψ is a strict extension of θ , then ϕ is a strict extension of θ , too.6. dimS = dimX.

Proof. There exists a continuous mapping f :βX → βZ such that f (x) = x for any x ∈ X. Let Z1 = Z ∩ f (βX),S = f −1(Z1) and g = f |S. The proof is completed. �

M.M. Choban and S.I. Nedev [5] proved: Let X be a subspace of some linear ordered space and Y be a reflexiveBanach space. Then for every closed-valued lower semi-continuous mapping θ :X → Y there exists a strong para-compact extension ψ :Z → Y , where Z is a subspace of some linear ordered space. This principle was successfullyapplied by I. Shishkov [23–25].


5. Principle of factorization

A mapping θ :X → Y admits a paracompact factorization if X is a completely regular space and there exist aparacompact Hausdorff space Z, a single-valued continuous mapping g :X → Z of X into Z and a lower semi-continuous mapping ϕ :Z → Y such that ϕ(g(x)) ⊆ θ(x) for any x ∈ X. If θ = ϕ ◦ g, then we say that θ admits astrong paracompact factorization (see [16,15]).

Proposition 5.1. For a set-valued mapping θ :X → Y the following assertions are equivalent:

1. θ admits a paracompact extension.2. For θ there exists a strong paracompact factorization.

Proof. Let ϕ :Z → Y be an extension of θ , Z be a paracompact space and g :X → Z be the embedding of X into Z.Then θ = ϕ ◦ g and the implication 1 → 2 is proved. Let X be a completely regular space, Z be a Hausdorff paracom-pact space, g :X → Z be a continuous single-valued mapping and ψ :Z → Y be a lower semi-continuous mappingsuch that θ(x) = ψ(g(x)) for any x ∈ X. There exists a continuous extension f :βX → βZ of the mapping g. We putS = f −1(Z) and ϕ(z) = ϕ(f (z)) for any z ∈ S. Then ϕ :S → Y is an extension of θ . The proof is completed. �6. Principle of equivalence of mappings

Definition 6.1. Two set-valued mappings ϕ :X → Y and ψ :X → Z are called equivalent if there exists a homeomor-phism g : Gr(ϕ) → Gr(ψ) such that ψ−1

X (x) = g(ϕ−1X (x)) for every point x ∈ X.

Theorem 6.2. Let ϕ :X → Y and ψ :X → Z be two equivalent set-valued mappings.

1. The mapping ϕ is lower semi-continuous if and only if the mapping ψ is lower semi-continuous.2. The mapping ϕ is compact-valued if and only if the mapping ψ is compact-valued.3. The mapping ϕ is single-valued if and only if the mapping ψ is single-valued.4. The mapping ϕ is compact-valued and upper semi-continuous if and only if the mapping ψ is compact-valued

and upper semi-continuous.5. For every selection λ of the mapping ϕ there exists a selection μ = e(λ) of the mapping ψ such that λ and μ are

equivalent. Moreover, e is a one-to-one correspondence between the set of selections of ϕ and the set of selectionsof ψ which preserves the relation of equivalence.

Proof. Let g : Gr(ϕ) → Gr(ψ) be a homeomorphism such that ψ−1X (x) = g(ϕ−1

X (x)) for any point x ∈ X. In this caseψ(x) = ψZ(g(ϕ−1

X (x))) and ϕ(x) = ϕY (g−1(ψ−1X (x))) for every point x ∈ X.

Assertions 2 and 3 are proved. Assertion 1 follows from Proposition 1.1. Assertion 4 follows from Assertion 2 andProposition 1.2.

Let λ :X → Y be a selection of the mapping ϕ. Then Gr(λ) ⊆ Gr(ϕ) and λX = ϕ|Gr(λ). For every point x ∈ X

we put μ(x) = ψz(g(λ−1X (x))). By construction, μ :X → Z is a selection of the mapping ψ and Gr(μ) = g(Gr(λ)).

In particular, the mappings λ and μ are equivalent. We assume that μ = e(λ). Assertion 5 is proved. The proof isfinished. �Corollary 6.3. Every two equivalent mappings have the same “nice” selections.

7. Metrizable families of sets

We say that ρ is a continuous pseudometric on a space X if the ball B(x,ρ, r) = {y ∈ X: ρ(x, y) < r} is open inX for any x ∈ X and every real number r .

Let ρ be a pseudometric on a space X. Then there exist a set X/ρ and a mapping πρ :X → X/ρ onto X/ρ suchthat π−1

ρ (πρ(x)) = {y ∈ X: ρ(x, y) = 0} for any point x ∈ X. The function ρ(πρ(x),πρ(y)) = ρ(x, y) is a metricon X/ρ. On X/ρ we consider the topology generated by the metric ρ. We put H(x,ρ) = {y ∈ X: ρ(x, y) = 0}, i.e.


H(x,ρ) = π−1ρ (πρ(x)). The mapping πρ is continuous if and only if the pseudometric ρ is continuous. If H is a

nonempty subset of X, then diam(H) = sup{ρ(x, y): x, y ∈ H } is the diameter of the set H relatively to ρ.

Definition 7.1. A family M of subsets of a space X is called metrizable if there exists a continuous pseudometric ρ

on X such that:

1. If L ∈ M and x ∈ L, then L ∩ H(x,ρ) = {x}.2. If L0 ∈ M, x0 ∈ L0 and U is an open subset of X such that x0 ∈ U , then there exist an open subset V of X and

ε > 0 such that L ∩ B(x,ρ, ε) ⊆ U provided L ∈M and x ∈ L ∩ V .

In the conditions of Definition 7.1 we say that the family M is metrizable by the continuous pseudometric ρ.

Proposition 7.2. Suppose that a family M of subsets of space X is metrizable by the continuous pseudometric ρ on X.

1. If L ∈M, then L is metrizable by the metric ρ on L.2. If X = ⋃{L ∈M: x ∈ L} for every x ∈ X, then the space X is metrizable by the metric ρ.

Proof. Follows immediately from Definition 7.1. �Definition 7.3. A family M of subsets of a space X is called complete metrizable if there exists a continuous pseudo-metric ρ on X such that:

1. The family M is metrizable by the pseudometric ρ.2. For every L ∈M the metric ρ on L is complete.

Proposition 7.4. Suppose that a family M of subsets of a space X is metrizable by a continuous pseudometric ρ1and ρ2 is a continuous pseudometric on X such that ρ2 is a complete metric on every L ∈ M. Then the family M iscomplete metrizable by the continuous pseudometric ρ = ρ1 + ρ2.

Proof. Obvious. �Example 7.5. Let ρ be a continuous pseudometric on a space X and r > 0. A subset L ⊆ X is r-discrete if ρ(x, y) � r

for every pair of distinct points x, y ∈ L. Denote by M(ρ, r) the family of all r-discrete subsets of X. Then M(ρ, r)

is a complete metrizable family of subsets of X.

Example 7.6. Let ρ be a continuous metric on a nonmetrizable space X and {F ⊆ X: |F | < 2} ⊆ M ⊆ {F ⊆ X: F isa compact subset}. Every subspace L ∈ M of X is complete metrizable by the metric ρ and the family M is notmetrizable.

Example 7.7. Let X be a metrizable space by the metric ρ. Then every family of subsets of X is metrizable by ρ. Thefamily C(ρ) = {Y ⊆ X: ρ is a complete metric on Y } is complete metrizable by ρ.

Example 7.8. Let X be a space and M = {{x}: x ∈ X} be the family of all singleton subsets. We put ρ(x, y) = 0 forany x, y ∈ X. Then M is complete metrizable by the continuous pseudometric ρ on X.

8. Metrizable set-valued mappings

Definition 8.1. A set-valued mapping θ :X → Y is called:

– metrizable if θ is equivalent to some mapping ϕ :X → Z into some metrizable space Z;– complete metrizable if θ is equivalent to some closed-valued mapping ϕ :X → Z into some complete metrizable

space Z;


– regular metrizable if θ is equivalent to some closed-valued mapping ϕ :X → Z into some complete metrizablespace Z such that ϕ admits a paracompact extension.

If M is a family of subsets of a space Y , θ :X → Y is a set-valued mapping and θ(x) ∈M for any x ∈ X, then wedenote θ :X → M and say that θ is a mapping of the space X into M.

Theorem 8.2. Let M be a family of subsets of a space Y , ρ be a continuous pseudometric on Y , θ :X → M be alower semi-continuous mapping and θ (x) = πρ(θ(x)) for any x ∈ X.

1. The mapping θ :X → Y/ρ is lower semi-continuous.2. If the family M is metrizable by the pseudometric ρ, then the mappings θ and θ are equivalent and, in particular,

θ is a metrizable mapping.3. If the family M is complete metrizable, then the mapping θ is complete metrizable.

Proof. Assertion 1 is obvious.For every point (x, y) ∈ X ×Y we put h(x, y) = (x,πρ(y)). Then h :X ×Y → X ×Y/ρ is a continuous mapping.Suppose that the family M is metrizable by the pseudometric ρ. In this case g = h|Gr(θ) : Gr(θ) → Gr(θ) is an

one-to-one continuous mapping. Moreover, θ−1X (x) = g(θ−1

X (x)) for every point x ∈ X.Fix (x0, y0) ∈ Gr(θ), an open subset U of X and an open subset W of Y such that x0 ∈ U and y0 ∈ W . It is obvious

that y0 ∈ θ(x0) ∈ M. There exist ε > 0 and an open subset V of Y such that y0 ∈ V and θ(x) ∈ B(y,ρ,2ε) ⊆ W

provided y ∈ θ(x) ∩ V . Let W1 = {y ∈ Y : ρ(y0, y) < ε} and U1 = U ∩ θ−1(V ∩ W1) = {x ∈ U : θ(x) ∩ V ∩ W1 �= ∅}.We affirm that Gr(θ) ∩ (U1 × W1) ⊆ U × W . Let x ∈ U1 and y ∈ θ(x) ∩ W1. Then x ∈ U and there exists some pointy′ ∈ θ(x) ∩ V ∩ W1. Thus ρ(y0, y) < ε, ρ(y0, y

′) < ε and ρ(y, y′) < 2ε. Hence y ∈ θ(x) ∩ B(y′, ρ,2ε) ⊆ W and(x, y) ∈ U ×W . Therefore the mapping g−1 : Gr(θ) → Gr(θ) is continuous and g is a homeomorphism. In particular,the mappings θ and θ are equivalent. The assertion 2 is proved. If the family M is complete metrizable by ρ and Z isthe completion of the metric space (Y/ρ, ρ), then θ :X → Z is a closed-valued mapping. Assertion 3 is proved. Theproof is completed. �9. Applications to topological groups

Let H be a subgroup of the topological group G. Consider the family MH = {x · H : x ∈ G} and the open contin-uous mapping pH :G → G/H , where p−1

H (pH (x)) = xH for any point x ∈ G. In particular, the set-valued mappingϕH :G/H → G, where ϕH = p−1

H , is lower semi-continuous.

Property 1. If H is a metrizable subgroup of G, then the family MH is metrizable.

The subgroup H of the group G is a uniform subgroup if for every neighborhood U of the identity e ∈ G thereexists a neighborhood V of e such that HV ⊆ UH (see [1]).

Every normal subgroup and every compact subgroup is uniform.

Property 2. Let H be a metrizable uniform subgroup of the group G. Then the set-valued mapping ϕH :G/H → G

is equivalent to a mapping which admits a strong metrizable factorization and a paracompact extension.

Property 3. Let H be a metrizable subgroup complete in the left uniformity of the group G. Then the family MH iscomplete metrizable by some continuous pseudometric ρ.

Property 4. Let H be a metrizable Cech complete normal subgroup of the group G. Then the family MH is completemetrizable.

A topological group G is almost metrizable if it contains a non-empty compact subset of countable character in G

(see [17]).


Property 5. Let H be a complete metrizable subgroup of an almost metrizable group G. Then the mappingϕH :G/H → G admits a paracompact extension and the family MH is complete metrizable.

Properties 1–5 were proved in [1,4]. These properties, the possibility of the approximation of compact group andthe well-known theorems about existence of selections permit to obtain extensive information concerning selectionsin the topological groups (see [1,2,8,18,19] and references).

We mention the following facts.

Corollary 9.1. Let H be a Cech complete subgroup of the group G, g :X → G/H be a single-valued continuousmapping of a space X into G/H and one of the following conditions is satisfied:

1. H is complete relatively to left uniformity and X is a paracompact space.2. H is a uniform subgroup and X is paracompact.3. H is a uniform subgroup which is complete relatively to the left uniformity.4. H is a normal subgroup.5. G is an almost metrizable group.

Then there exist an upper semi-continuous compact-valued mapping ψ :X → G and a lower semi-continuouscompact-valued mapping ϕ :X → G such that ϕ(x) ⊆ ψ(x) ⊆ p−1

H (g(x)) for any x ∈ X. If dimX = 0, then ψ is asingle-valued mapping.

10. On metrizable families of locally convex spaces

Let E be a linear topological space and q :E → R be a continuous pseudonorm:

1. q(αx) = |α| · q(x) for all α ∈ R and x ∈ E;2. q(x + y) � q(x) + q(y) for all x, y ∈ E;3. The set {x: q(x) < 1} is open in E.

A function q :E → R is a continuous pseudoquasinorm if:

1. q(0) = 0.2. q(x + y) � q(x) + q(y).3. q(−x) = q(x).4. If {αn ∈ R: n ∈ N} and limαn = 0, then limq(αnx) = 0 for every x ∈ E.5. If {xn ∈ E: n ∈ N} and limq(xn) = 0, then lim(αxn) = 0 for any α ∈ R.6. The set {x ∈ E: q(x) < r} is open in E for every r ∈ R.

Definition 10.1. Let α � 0. A family M of subsets of a linear topological space E is a complete metrizable α-para-convex family of the space E if there exists a continuous pseudonorm q on E such that:

1. The family M is complete metrizable by the continuous pseudometric ρ(x, y) = q(x − y).2. If r > 0, H ∈ M, a ∈ E, D = {x ∈ E: ρ(a, x) < r} and b ∈ conv(H ∩D), then ρ(b,H) = inf{ρ(b, s): x ∈ H } �

α · r .

For α = 0 we say that M is a complete metrizable convex family of subsets of the space E.

Theorem 10.2. Let 0 � α < 1 and M be a complete metrizable α-paraconvex family of subsets of a linear topologicalspace E. Then for every lower semi-continuous set-valued mapping θ :X → M of a Hausdorff paracompact spaceX into M there exists a continuous single-valued selection.

Proof. Suppose that q is a continuous pseudonorm on E with the properties from Definition 10.1.


Consider the natural projection pq :E → E/q . Then q(pq(x)) = q(x) is a norm on E/q . Let B be the completionof the normed space E/q . In this case {pq(H): H ∈ M} is a family of closed α-paraconvex subsets of the Banachspace B . If θ (x) = pq(θ(x)) for any x ∈ X, then in virtue of Michael’s selection theorem [10] for the mapping θ thereexists a continuous single-valued selection. By virtue of Theorems 8.2 and 6.2 the proof is finished. �

Example 10.3. Let L be a linear subspace of the linear topological space E. We say that L is B-embedded in E ifthere exists a continuous pseudonorm q on E which is a complete norm on L. In this case L is a Banach subspaceof E and M = {x + L: x ∈ E} is a complete metrizable convex family of subsets of E. Moreover, the set-valuedmapping p−1

L :E/L → E is regular metrizable. Hence there exists a subset X of E such that g = pL|X :X → E/L isa homeomorphism into E/L. In particular, the spaces E and L × E/L are homeomorphic.

Example 10.4. Let E be a linear topological space. We say that a continuous inner product is defined on a space E iffor any x, y ∈ E the real number q(x, y) with the next properties is defined:

P1. q(x + y, z) = q(x, z) + q(y, z) for all x, y, z ∈ E.P2. q(αx, y) = αq(x, y) for all α ∈ R and x, y ∈ E.P3. q(y, x) = q(x, y) for all x, y ∈ E.P4. q(x) = q(x, x) is a continuous pseudonorm on E.

Fix a continuous inner product q on E. Let L = {x ∈ L: q(x, x) = 0}. Then E/L is a pre-Hilbert space. In thiscase as in [21] we can define the β-star subsets of the space E relatively to the inner product q . Thus we can definethe complete metrizable β-star families of subsets of E and may obtain the corresponding selection results.

Example 10.5. A family M of subsets of a linear topological space E is called a uniformly locally convex metrizablefamily of subsets if on E there exists a continuous linear invariant pseudometric ρ such that:

1. M is complete metrizable by the pseudometric ρ.2. Every H ∈ M is a convex subset of E.3. For every ε > 0 there exists a δ > 0 for which diam(conv(B(x,ρ, δ) ∩ H)) < ε for all x ∈ H ∈M.4. For every x ∈ E, α ∈ R and ε > 0 there exists a δ > 0 such that ρ(αx,βy) < ε provided ρ(x, y) < δ and

|α − β| < δ.5. ρ(x + z, y + z) = ρ(x, y) for all x, y, z ∈ E.

A pseudometric ρ on E is linearly invariant if and only if q(x − y) = ρ(x, y) for some pseudoquasinorm q on E.In this case E/ρ is a metrizable linear space, ρ is a metric on E/ρ and Mρ = {πρ(H): H ∈ M} is a uniformlocally convex family of convex complete subsets of E/ρ in the sense of P.V. Semenov. Recently P.V. Semenov[22] proved a selection theorem for lower semi-continuous set-valued mappings into a family of uniformly convexcomplete subsets of a linear metrizable space. The assertion of this theorem remains true for a lower semi-continuousset-valued mapping into a uniformly locally convex metrizable family of an arbitrary linear topological space.

Example 10.6. Let L be a linear subspace of a linear topological space E. The subspace L is called locally uniformlyconvex in E if for every neighborhood U of the zero 0 in E there exists a neighborhood V of 0 in E such thatconv(L ∩ (x + V )) ⊆ U for every x ∈ L. If L is a complete metrizable subspace of E, then:

1. The family ML = {x + L: x ∈ E} is a uniformly locally convex metrizable family of subsets of E.2. There exists a closed subspace X of E such that g = pL|X :X → E/L is a homeomorphism of X into E/L.3. For every lower semi-continuous mapping ϕ :X → ML of an arbitrary space X there exists a continuous single-

valued selection.


11. Final conclusions

Let μ be a class of metric spaces, S be a class of topological spaces and P be a property of subsets of the spacesfrom μ.

For every space X ∈ μ denote by exp(X,P) the family of all nonempty subsets of X with the property P .In these conditions we may examine the metrizable families with the property P in the class of all topological

spaces.

Definition 11.1. A family M of subsets of a space X is called a metrizable family with the property P if there exist acontinuous pseudometric ρ on X and a metric space Z ∈ μ for which:

1. The family M is metrizable by the pseudometric ρ.2. X/ρ is a subspace of the space Z.3. {πρ(H): H ∈ M} ⊆ exp(Z,P).

Theorems 6.2 and 8.2 yield

Corollary 11.2. Let A and B be two properties of set-valued mappings which are invariant by the relation of theequivalence. The following assertions are equivalent:

1. For every lower semi-continuous set-valued mapping θ :X → exp(Y,P) with the property A of a space X ∈ S

into a space Y ∈ μ there exists a selection ϕ :X → Y with the property B.2. For every lower semi-continuous set-valued mapping θ :X → M with the property A of a space X ∈ S into a

metrizable family M with the property P of subsets of a space Y there exists a selection ψ :X → Y with theproperty B.

The results mentioned in the preceding sections are special cases of Corollary 11.2. In this way many resultsabout existence of “nice” selection can be extended and generalized for lower semi-continuous set-valued mapping invarious classes of metrizable spaces.

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On selections and classes of spaces

Mitrofan M. Choban, Ekaterina P. Mihaylova ∗, Stoyan Y. Nedev

Bulgarian Academy of Sciences, Institute of Mathematics, Acad. G. Bonchev str. Bl. 8, Sofia, Bulgaria

Received 31 October 2006; received in revised form 31 May 2007; accepted 31 May 2007

Abstract

Strong paracompactness, Lindelöf number and degree of compactness are characterized in terms of selections of set-valuedmappings.© 2007 Elsevier B.V. All rights reserved.

MSC: 54C60; 54C65; 54D20; 54D30

Keywords: Set-valued mapping; Selection; Compactness; Strong paracompactness; Lindelöf number; Compactness degree

1. Introduction

In the early 50’s E. Michael characterized paracompactness, countable paracompactness, normality, collection-wise normality and some other topological invariants by means of selection theorems. This characterizations showeduseful in different extent thus stimulating many authors to augment the list of topological invariants characterizedvia selections (see [1–4,7,8,10,11]). In the present paper we add to that list the strong paracompactness, the Lindelöfnumber and the degree of compactness. More precisely, we characterize the above invariants in terms of selections ofset-valued mappings into discrete topological spaces.

Our terminology comes, as a rule, from [5,7,11].A family γ of subsets of a space X is star-finite (star-countable) if for every element Γ ∈ γ the set {L ∈ γ :

L ∩ Γ �= ∅} is finite (countable).A topological space X is called strongly paracompact or hypocompact if X is Hausdorff and every open cover of

X has a star-finite open refinement.A shrinking of a cover ξ = {Uα: α ∈ A} of the space X is a cover γ = {Vα: α ∈ A} such that Vα ⊆ Uα for

every α ∈ A (see [5,6]). The operation of shrinking preserves the properties of local finiteness, star-finiteness, star-countableness.

The cardinal number l(X) = min{m: every open cover of X has an open refinement of cardinality � m} is theLindelöf number of X.

* Corresponding author.E-mail addresses: [email protected] (M.M. Choban), [email protected] (E.P. Mihaylova), [email protected] (S.Y. Nedev).


798 M.M. Choban et al. / Topology and its Applications 155 (2008) 797–804

The cardinal number k(X) = min{m: every open cover of X has an open refinement of cardinality < m} is thedegree of compactness of X.

Denote by τ+ the least cardinal number larger than the cardinal number τ . It is obvious that l(X) � k(X) � l(X)+.

2. Strong paracompactness

Let X and Y be nonempty topological spaces. A set-valued mapping θ : X → Y assigns to every x ∈ X a non-empty subset θ(x) of Y . If φ,ψ : X → Y are set-valued mappings and φ(x) ⊆ ψ(x) for every x ∈ X, then φ is calleda selection of ψ .

Let θ : X → Y be a set-valued mapping and let A ⊆ X and B ⊆ Y . The set θ−1(B) = {x ∈ X: θ(x) ∩ B �= ∅} is theinverse image of B , θ(A) = θ1(A) = ⋃{θ(x): x ∈ A} is the image of the set A and θn+1(A) = θ(θ−1(θn(A))) is the(n + 1)-image of the set A. The set θ∞(A) = ⋃{θn(A): n ∈ N} is the largest image of the set A.

A set-valued mapping θ : X → Y is called lower (upper) semi-continuous if for every open (closed) subset H of Y

the set θ−1(H) is open (closed) in X.A family {Hα: α ∈ A} of subsets of a space X is closure-preserving if

⋃{clXHβ : β ∈ B} = clX(⋃{Hβ : β ∈ B})

for every B ⊆ A (see [9]).Let X be a space, γ = {Uα: α ∈ Y } be a cover of X, Y being endowed with the discrete topology and θγ (x) =

{α ∈ Y : x ∈ Uα} for every x ∈ X. The mapping θγ :X → Y is lower semi-continuous if and only if the cover γ isopen. The mapping θγ is upper semi-continuous if and only if the set

⋃{Uα: α ∈ B} is closed in X for every subsetB of Y . The mapping θγ is compact-valued if and only if the cover γ is point-finite. If B ⊆ Y and λ = {Vβ : β ∈ B} isa cover of the space X, then θλ : X → B ⊆ Y is a selection of θγ if and only if λ is a shrinking of γ (one may assumethat Vα = ∅ for every α ∈ Y \ B).

Many covering properties of spaces may be reformulated in terms of selections for lower semi-continuous map-pings into discrete spaces. It is desirable to characterize these notions (including the strong paracompactness) in termsof selections for lower semi-continuous mappings into complete metric spaces.

Lemma 1. For a T1-space X the following are equivalent:

(1) X is strongly paracompact.(2) For every open cover of X there exists a closed closure-preserving star-countable refinement.

Proof. (1) → (2) Implication (1) → (2) is well known (see [5]).(2) → (1) Let F and Φ be two disjoint closed subsets of X and the closed closure-preserving cover L is a refine-

ment of the open cover {X \ F,X \ Φ}. Put U = X \ ⋃{L ∈ L: L ∩ F = ∅} and V = X \ ⋃{L ∈ L: L ∩ Φ = ∅}. Thesets U,V are open in X, F ⊆ U , Φ ⊆ V and U ∩ V = ∅. Thus X is normal and regular.

Let γ = {Uα: α ∈ A} be an open cover of X. There exist a closed closure-preserving star-countable cover ξ ={Hz: z ∈ Z} of X and a single-valued mapping g : Z → A such that Hz ⊆ Ug(z) for every z ∈ Z. For every S ⊆ X

put ξ(S) = ⋃{H ∈ ξ : H ∩ S �= ∅}. For every z ∈ Z put X1(z) = Hz, Xn+1(z) = ξ(Xn(z)) for every n ∈ N andX(z) = ⋃{Xn(z): n ∈ N}. By construction, every X(z) is an open-and-closed subset of X.

Put Bz = {b ∈ Z: Hb ∩X(z) �= ∅} = {b ∈ Z: Hb ⊆ X(z)} for z ∈ Z. Obviously Bz is countable for every z ∈ Z. Forevery z, t ∈ Z either X(z) = X(t) or X(z)∩X(t) = ∅. Fix an open subset Vz of X for which Hz ⊆ Vz ⊆ X(z)∩Ug(z).Then μ = {Vz: z ∈ Z} is a star-countable open refinement of γ . By virtue of [5], Theorem 5.3.10, the space X isstrongly paracompact. �Theorem 2. For a T1-space X the following are equivalent:

(1) X is strongly paracompact.(2) For every lower semi-continuous set-valued mapping θ : X → Y into a discrete space Y there exist a discrete

space Z, a single-valued mapping g : Z → Y , a lower semi-continuous mapping φ : X → Z and an uppersemi-continuous finite-valued mapping ψ : X → Z such that g(φ(x)) ⊆ g(ψ(x)) ⊆ θ(x) and the set ψ∞(x)

is countable for every x ∈ X.

M.M. Choban et al. / Topology and its Applications 155 (2008) 797–804 799

(3) For every lower semi-continuous mapping θ : X → Y into a discrete space Y there exist a discrete space Z, asingle-valued mapping g : Z → Y and an upper semi-continuous mapping ψ : X → Z such that g(ψ(x)) ⊆ θ(x)

and the set ψ∞(x) is countable for every x ∈ X.(4) X is regular and for every lower semi-continuous mapping θ : X → Y into a discrete space Y there exist a

discrete space Z, a single-valued mapping g : Z → Y and a lower semi-continuous mapping φ : X → Z suchthat g(φ(x)) ⊆ θ(x) and the set φ∞(x) is countable for every x ∈ X.

Proof. Our proof goes as follows: (1) → (2) → (3) → (1) → (2) → (4) → (1).(1) → (2) The set θ−1(y) is open in X and {θ−1(y): y ∈ Y } is an open cover of X. There exist a discrete space

Z, a star-finite open cover {Uz: z ∈ Z} of X and a single-valued mapping g : Z → Y such that clXUz ⊆ θ−1(g(z))

for every z ∈ Z. It is obvious that {clXUz: z ∈ Z} is a closed locally finite cover of X. There exists an open cover{Vz: z ∈ Z} of X such that clXVz ⊆ Uz for every z ∈ Z. The cover {clXVz: z ∈ Z} is star-finite.

Put φ(x) = {z ∈ Z: x ∈ Vz} and ψ(x) = {z ∈ Z: x ∈ clXVz} for every x ∈ X. It is obvious that φ is a lower semi-continuous finite-valued mapping, ψ is an upper semi-continuous finite-valued mapping and g(φ(x)) ⊆ g(ψ(x)) ⊆θ(x) for every x ∈ X.

Let z ∈ Z. Then ψ−1(z) = clXVz and ψ(ψ−1(z)) = {y ∈ Z: clXVz ∩ clXVy �= ∅}. Thus ψ(ψ−1(z)) is a finite setfor every z ∈ Z. Therefore the sets ψn(x) are finite and the set ψ∞(x) is countable for every x ∈ X.

(2) → (3) The implication (2) → (3) is obvious.(3) → (1) and, simultaneously, (2) → (4).Let F and Φ be two non-empty disjoint closed subsets of X. Define the mapping θ : X → {0,1}, by θ−1(0) = X\Φ

and θ−1(1) = X \F . θ is lower semi-continuous. Let ψ : X → {0,1} be an upper semi-continuous selection of θ . Thesets U = X \ ψ−1(0) and V = X \ ψ−1(1) are open in X, U ∩ V = ∅, Φ ⊆ U and F ⊆ V . Therefore X is a normalT1-space and hence regular.

In particular implication (2) → (4) has been established as well.Let now γ = {Uα: α ∈ A} be an open cover of X. Endow A with the discrete topology and put θγ (x) =

{α ∈ A: x ∈ Uα}, x ∈ X. Since θ−1γ (α) = Uα for every α ∈ A, the mapping θγ : X → A is lower semi-continuous. Let

Z be a discrete space, g : Z → A be a single-valued mapping and ψ : X → Z be an upper semi-continuous mappingsuch that g(ψ(x)) ⊆ θγ (x) and the set ψ∞(x) is countable for every x ∈ X. Then λ = {Hz = ψ−1(z): z ∈ Z} is astar-countable cover of X, Hz ⊆ Ug(z) for every z ∈ Z and the set

⋃{Hz: z ∈ B} = ψ−1(B) is closed in X for everysubset B ⊆ Z. By virtue of Lemma 1 the space X is strongly paracompact.

(4) → (1) Let φ : X → Z be a lower semi-continuous mapping, g(φ(x)) ⊆ θγ (x) with φ∞(x) countable forevery x ∈ X. Then {φ−1(z): z ∈ Z} is a star-countable open refinement of γ . [5], Theorem 5.3.10, completes theproof. �Remark 3. Let φ,ψ and g be as in Theorem 2, φ1(x) = g(φ(x)) and ψ1(x) = g(ψ(x)) for every x ∈ X. Then φ1 isa lower semi-continuous selection of θ and ψ1 is an upper semi-continuous selections of θ . Note that the set φ∞

1 (x)

needs not be countable.

For a space X put ω(X) = ⋃{U : U is open in X and dimU = 0} and let cω(X) = X \ω(X) be the cozero-dimen-sional kernel of X.

Let us recall that a normal space X satisfies the inequality dimX � n (n a non negative integer) if and only if everyfinite open cover of the space X has an open shrinking of order � n (see [5,6]). In Propositions 4 and 5 that follow wedo apply the above definition of dim to (open) subsets of normal spaces as well, just for simplicity of exposition. Thisdeviation from the standard “dimX = dimβX for a completely regular space X” does not affect neither validity norgenerality of the cited propositions.

Observation 1. By definition, dimX = 0 (that is X is not empty and dimX � 0) if and only if every finite open coverof the space X has an open discrete shrinking. If dimX = 0, F is a closed subset of X, U is an open subset of X,F ⊆ U and {V,W } is a discrete shrinking of the open cover {U,X \ F }, then the set V is open-and-closed in X andF ⊆ V ⊆ U (see [5], Theorem 7.1.10). Moreover, if X is a regular space, then ω(X) = ⋃{U : U is open-and-closedin X and dimU = 0}.


Proposition 4. For a Hausdorff space X the following are equivalent:

(1) X is paracompact and cω(X) is Lindelöf.(2) X is strongly paracompact and cω(X) is Lindelöf.(3) X is regular and for every open cover ξ = {Uα: α ∈ A} of X there exists an open star-countable cover γ =

{Vα: α ∈ A} such that Vα ⊆ Uα for every α ∈ A.

Proof. (1) → (2) → (3) Suppose X is paracompact and cω(X) is Lindelöf. Let ξ = {Uα: α ∈ A} be an open coverof X. Since every open cover of a paracompactum has an open locally finite shrinking ([5], Remark 5.1.7), onemay assume that ξ is locally finite. There exists a countable subset B ⊆ A and an open subset W of X such thatcω(X) ⊆ W ⊆ clXW ⊆ ⋃{Uα: α ∈ B}. Let F = clXW \ W , H = X \ ⋃{Uα: α ∈ B} and Y = X \ W . AssumeY is nonempty. Since Y is closed in X it is paracompact. By virtue of Observation 1, for every y ∈ ω(X) thereexists an open-and-closed subset Uy of X such that y ∈ Uy and dimUy = 0. In virtue of the Locally Finite SumTheorem for dim ([6], Theorem 3.1.10), dimY = 0. Obviously, the same reasons as for Y , dimP = 0 for everyclosed nonempty subset P of X such that P ⊆ ω(X). Thus there exists an open-and-closed subset G of Y such thatH ⊆ G ⊆ Y \ F = X \ clXW ⊆ Y (see [5], Theorem 7.1.10). Obviously G is closed in X, too. Since G is open inY \ F = X \ clXW and X \ clXW is open in X, the set G is open-and-closed in X.

Therefore the set V = X \ G is an open-and-closed subset of X with cω(X) ⊆ V ⊆ ⋃{Uα: α ∈ B}. Sincedim(X \ V ) = 0 (G = X \ V is closed in X, it is a subset of ω(X) and one may assume H ⊆ G is nonempty,otherwise the proof is obvious), there exists a discrete family of open-and-closed subsets {Wα: α ∈ A} of X such thatX \ V = ⋃{Wα: α ∈ A} and Wα ⊆ Uα for every α ∈ A (see [5], Theorem 7.2.4). Let Hα = V ∩ Uα for α ∈ B andHα = ∅ for α ∈ A \ B . The family {Hα: α ∈ A} is open and star-countable. By construction, V = ⋃{Hα: α ∈ A} andHα ⊆ Uα for every α ∈ A. Put Vα = Hα ∪ Wα for every α ∈ A. (If Y is empty, put Vα = Uα for α ∈ B and Vα = ∅ forα ∈ A \ B .)

The family γ = {Vα: α ∈ A} is an open star-countable cover and Vα ⊆ Uα for every α ∈ A.(3) → (1) In virtue of [5], Theorem 5.3.10, X is strongly paracompact and hence paracompact and normal. Let

{Pμ: μ ∈ M} be a discrete family of closed subsets of X and dimPμ � 1 for every μ ∈ M . For every μ ∈ M fixtwo disjoint closed subsets Ψμ,Φμ of Pμ such that there is no open subset H for which Ψμ ⊆ H ⊆ X \ Φμ andH ∩ Pμ = Pμ ∩ clXH . Suppose that 0 /∈ M and A = M ∪ {0}. Put U0 = X \ ⋃{Φμ: μ ∈ M} and Uμ = X \ (Ψμ ∪⋃{Pν : ν ∈ M \ {μ}}) for every μ ∈ M . By construction, ξ = {Uα: α ∈ A} is an open cover of X. If χ = {Hα: α ∈ A}is an open cover of X and Hα ⊆ Uα,α ∈ A, then:

(1) H0 ∩ Hμ �= ∅ for every μ ∈ M .(2) If the set M is uncountable, then the cover χ is not star-countable.(3) If the set M is infinite, then the cover χ is not star-finite.

If the subspace cω(X) is not Lindelöf, then we can suppose that M is an uncountable set and⋃{Pμ: μ ∈ M} ⊆

cω(X).Let’s be more specific, especially for the cases (2) and (3) above. To establish the existence of a family {Pμ: μ ∈ M}

with the required properties one may pass through the following observations:

Observation 2. The closure of every non-empty open subset of Y = cω(X) is not zero-dimensional (in sense dim).Assume the opposite and let U be open in X with U ∩ Y �= ∅ and clX(U ∩ Y) = clY (U ∩ Y) be zero-dimensional.Notice that since clX(U ∩ Y) is a closed subset of a paracompactum, it is paracompact and normal. Therefore everyclosed subset of clX(U ∩ Y) is zero-dimensional too. Fix x0 ∈ U ∩ Y . Take an Fσ open (in X) neighborhood V ofx0 with clXV ⊆ U . We put Z = clX V and Z1 = Z ∩ cω(X). It is obvious that dimZ1 = 0. As Fσ -set in X, V isa normal subspace ([6], Lemma 3.1.22, or [5], Exercise 2.1.E). Now the Countable Sum Theorem for dim impliesdimV � dimZ. Moreover, every closed in X subset P (see the proof of the previous implications) of Z \ Z1 ⊆ ω(X)

is zero-dimensional. In virtue of Dowker’s Lemma (see for instance [6], Lemma 3.1.6) dimZ = 0. Hence dimV = 0and x0 ∈ V ⊆ ω(X), a contradiction.


Observation 3. Let ζ be an open cover of Y having no countable open refinement. Assume {Gμ: μ ∈ M} is a locallyfinite open refinement of ζ . Pick a point xμ ∈ Gμ. The set {xμ: μ ∈ M} is discrete one (after identifying the coincidingxμ’s; note, this identification does not affect the cardinality of M which is always uncountable for case (2) (infinitefor case (3))), so, by the collection-wise normality of Y there exists a discrete family of open sets {Dμ: μ ∈ M} withxμ ∈ Dμ ⊆ clY Dμ ⊆ Gμ for every μ ∈ M . Now put Pμ = clY Dμ.

At last, since Pμ is zero-dimensional for no μ ∈ M , there are O1μ and O2

μ open subsets of Pμ with Pμ = O1μ ∪ O2

μ

and this last two-element cover of Pμ has no open-and-closed shrinking. Now we let Ψμ = Pμ \ O1μ and Φμ =

Pμ \ O2μ. �

The following proposition can be proved in a similar way.

Proposition 5. For a Hausdorff space X the following are equivalent:

(1) X is paracompact and cω(X) is compact.(2) For every open cover ξ = {Uα: α ∈ A} of the space X there exists an open star-finite cover γ = {Vα: α ∈ A} such

that Vα ⊆ Uα for every α ∈ A.

Proof. (1) → (2) Suppose that X is paracompact and cω(X) is compact. Let ξ = {Uα: α ∈ A} be an open coverof the space X. There exists a finite subset B ⊆ A and an open-and-closed subset V of X such that cω(X) ⊆ V ⊆⋃{Uα: α ∈ B}. The open cover γ = {Vα: α ∈ A}, constructed as in the proof of the previous proposition, is an openstar-finite cover and Vα ⊆ Uα for every α ∈ A.

(2) → (1) Suppose that X is paracompact and cω(X) is not compact. There exists an infinite discrete family{Pμ: μ ∈ M} of closed subsets of cω(X) such that dimPμ > 0 for every μ ∈ M . It follows, from the proof of theprevious proposition (case (3)), that there exists an open cover ξ = {Uα: α ∈ A} of X without open star-finite shrinkingwhich contradicts 2. �Theorem 6. For a T1-space X the following are equivalent:

(1) X is strongly paracompact and cω(X) is Lindelöf.(2) For every lower semi-continuous set-valued mapping θ : X → Y into a discrete space Y there exist a lower

semi-continuous mapping φ : X → Y and an upper semi-continuous finite-valued mapping ψ : X → Y such thatφ(x) ⊆ ψ(x) ⊆ θ(x) and the set ψ∞(x) is countable for every x ∈ X.

(3) For every lower semi-continuous mapping θ : X → Y into a discrete space Y there exists an upper semi-continuous mapping ψ : X → Y such that ψ(x) ⊆ θ(x) and ψ∞(x) is countable for every x ∈ X.

(4) X is regular and for every lower semi-continuous mapping θ : X → Y into a discrete space Y there exists a lowersemi-continuous mapping φ : X → Y such that φ(x) ⊆ θ(x) and φ∞(x) is countable for every x ∈ X.

Proof. (1) → (2) Let X be strongly paracompact, cω(X) be Lindelöf and θ : X → Y be a lower semi-continuousmapping into a discrete space Y . The set θ−1(y) is open in X and {θ−1(y): y ∈ Y } is an open cover of X. ByProposition 4, there exists a locally finite star-countable open cover {Uy : y ∈ Y } such that clXUy ⊆ θ−1(y) for everyy ∈ Y . It is obvious that {clXUy : y ∈ Y } is a closed locally finite cover of X. There exists an open cover {Vy : y ∈ Y }of X such that clXVy ⊆ Uy for every y ∈ Y . Note that the cover {clXVy : y ∈ Y } is star-countable.

Put φ(x) = {y ∈ Y : x ∈ Vy} and ψ(x) = {y ∈ Y : x ∈ clXVy} for every x ∈ X. It is obvious that φ is a lower semi-continuous finite-valued mapping, ψ is an upper semi-continuous finite-valued mapping and φ(x) ⊆ ψ(x) ⊆ θ(x) forevery x ∈ X.

Let y ∈ Y . Then ψ−1(y) = clXVy and ψ(ψ−1(y)) = {z ∈ Y : clXVy ∩ clXVz �= ∅}. Thus ψ(ψ−1(y)) is a countableset for every y ∈ Y . Therefore the sets ψn(x) and the set ψ∞(x) are countable for every x ∈ X.

(2) → (3), (2) → (4) Implications (2) → (3) and (2) → (4) are obvious.(3) → (1) and, simultaneously, (4) → (1).As in the proof of Theorem 2 we show that X is normal and thus regular. Let γ = {Uα: α ∈ A} be an open cover

of X. Endow A with the discrete topology and put θγ (x) = {α ∈ A: x ∈ Uα}. Since θ−1γ (α) = Uα for every α ∈ A,


the mapping θγ is lower semi-continuous. Let ψ : X → A be an upper semi-continuous selection of θγ with ψ∞(x)

countable for every x ∈ X. The family λ = {Hα = ψ−1(α): α ∈ A} is a star-countable cover of X, Hα ⊆ Uα for everyα ∈ A and, for every B ⊆ A the set

⋃{Hα: α ∈ B} = ψ−1(B) is closed in X . By virtue of Lemma 1 the space X isstrongly paracompact. As in the proof of Lemma 1 one can construct an open star-countable cover ξ = {Vα: α ∈ A}with Hα ⊆ Vα ⊆ Uα for every α ∈ A.

Let now φ : X → A be a lower semi-continuous mapping, φ(x) ⊆ θγ (x) with φ∞(x) countable for every x ∈ X.Then {φ−1(α): α ∈ A} is a star-countable open refinement of γ and Proposition 4 completes the proof. �

The following example for Y = [0,1] was suggested by the referee.

Example 7. Let Y be a compact non-empty space and Z be a non-empty discrete space. The space X = Y × Z hasthe following properties:

(1) X is strongly paracompact.(2) If dimY = 0, then cω(X) is empty.(3) If dimY > 0, then cω(X) = X.(4) If dimY > 0, then cω(X) is Lindelöf if and only if the set Z is countable.(5) If dimY > 0, then cω(X) is compact if and only if the set Z is finite.

3. On Lindelöf number and degree of compactness of spaces

In what follows singled-valued selection is not assumed to be continuous.

Theorem 8. For a regular space X and a cardinal number τ the following are equivalent:

(1) l(X) � τ .(2) For every lower semi-continuous closed-valued mapping θ : X → Y into a complete metrizable space Y there

exists a lower semi-continuous closed-valued selection ϕ : X → Y such that l(ϕ(X)) � τ .(3) For every lower semi-continuous set-valued mapping θ : X → Y into a complete metrizable space Y there exists

a single-valued mapping g : X → Y such that l(g(X)) � τ and g(x) ∈ clY θ(x) for every x ∈ X.(4) For every lower semi-continuous set-valued mapping θ : X → Y into a discrete space Y there exists a single-

valued selection g : X → Y such that |g(X)| � τ .(5) Every open cover of X has a refinement of cardinality � τ .

Proof. (1) → (2) Fix a cardinal number τ . Let l(X) � τ and θ : X → Y be a lower semi-continuous set-valuedmapping into a complete metric space (Y,ρ). It is well known that θ : X → Y , where θ(x) = clY θ(x), is lowersemi-continuous and closed-valued.

Moreover, θ−1(U) = θ−1(U) for every open subset U of Y . There exist a sequence γ = {γn = {Uα: α ∈ An}:n ∈ N} of open covers of X, a sequence λ = {λn = {Vα: α ∈ An}: n ∈ N} of families of open subsets of Y and asequence π = {πn :An+1 → An: n ∈ N} of mappings satisfying the following conditions for every n ∈ N and α ∈ An:

(1)⋃{Uβ : β ∈ π−1

n (α)} = Uα ⊆ clXUα ⊆ θ−1(Vα).(2)

⋃{clY Vβ : β ∈ π−1n (α)} ⊆ Vα .

(3) diamVα < 2−n and Vα �= ∅.(4) |An| � τ .

Assume that ρ(x, y) < 2−2 for every x, y ∈ Y . We construct the above staff by induction (see [4] or [10]). PutA1 = {1}, V1 = Y and U1 = X. Suppose that n � 1 and the objects γn,λn,πn−1 have already been constructed.Fix α ∈ An. For every y ∈ Vα fix an open subset Vy such that y ∈ Vy ⊆ clY Vy ⊆ Vα and diamVy � 2−n−2. Byconstruction, {θ−1(Vy): y ∈ Vα} is an open cover of the set clXUα . Since l(X) � τ and X is regular there exist a setAα , a family {Wβ : β ∈ Aα} of open subsets of X and a single valued mapping h : Aα → Vα , such that | Aα |� τ ,clXUα ⊆ ⋃{Wβ : β ∈ Aα} and clXWβ ⊆ θ−1(Vh(β)) for every β ∈ Aα . Assume that Aη ∩ Aβ = ∅ for every η,β ∈


Aα,η �= β . Let Vβ = Vh(β),Uβ = Uα ∩ Wβ,An+1 = ⋃{Aα: α ∈ An},π−1n (α) = Aα . The objects γn+1, λn+1,πn have

thus been constructed.Fix x ∈ X. Let An(x) = {α ∈ An: x ∈ Uα}. If α = (αn ∈ An(x): n ∈ N) and πn(αn+1) = αn for every n ∈ N, then

K(α) = ⋂{Vαn : n ∈ N} is a single-point set and K(α) ⊆ θ(x). Moreover, {Vαn : n ∈ N} is a base of the space Y at thepoint K(α). Put φ(x) = ⋃{K(α): α = (αn ∈ An(x): n ∈ N) and πn(αn+1) = αn for every n ∈ N} and ϕ(x) = clY φ(x)

for every x ∈ X. By construction, φ(x) ⊆ ϕ(x) ⊆ θ(x) for every x ∈ X and ϕ−1(H) = φ−1(H) for every open H ⊆ Y .Fix n ∈ N and β ∈ An. We claim that Uβ ⊆ ϕ−1(Vβ). If x ∈ Uβ , then there exists a sequence α = (αm ∈ Am(x): m ∈

N), αn = β,αm = πm(αm+1) for every m ∈ N. Then K(α) ⊆ ϕ(x) ∩ Vβ and x ∈ ϕ−1(Vβ). Therefore Uβ ⊆ ϕ−1(Vβ).Let H be an open subset of Y and x ∈ ϕ−1(H). There exists a sequence α = (αm ∈ Am(x): m ∈ N) such that

αm = πm(αm+1) for every m ∈ N and K(α) ⊆ ϕ(x) ∩ H . Since {Vαn : n ∈ N} is a base of the space Y at the pointK(α) there exists n ∈ N such that K(α) ⊆ Vαn ⊆ H and x ∈ Uαn ⊆ ϕ−1(Vαn) ⊆ ϕ−1(H). Thus the set ϕ−1(H) isopen in X.

Hence ϕ is a lower semi-continuous selection of θ . Since {Vα: α ∈ ⋃{An: n ∈ N}} is a base of the space Y at everypoint y ∈ ϕ(X), the weight of the subspace ϕ(X) is � τ . Hence l(ϕ(X)) � τ .

(2) → (3) For every point x ∈ X fix a point g(x) ∈ ϕ(x). Then g : X → Y is a single-valued selection of themapping θ and l(g(X)) � l(ϕ(X)) � τ .

(3) → (4) The implication (3) → (4) is obvious.(4) → (5) Let γ = {Uα: α ∈ A} be an open cover of X. Endow A with the discrete topology. The set-valued

mapping θγ :X → A, where θγ (x) = {α ∈ A: x ∈ Uα}, is lower semi-continuous. Let g : X → A be a selection of θγ

and |g(X)| = l(g(X)) � τ . Then {g−1(α): α ∈ g(X)} is a refinement of γ of cardinality � τ .(5) → (1) Implication (5) → (1) is obvious. �The proof of the following lemma is not difficult.

Lemma 9. Let Z be a subspace of a metric space (Y,ρ) and τ be an uncountable non-sequential cardinal number.The following are equivalent:

(1) k(Z) � τ .(2) w(Z) < τ (w(Z) being the weight of the space Z).(3) k(clY (Z)) � τ . �Theorem 10. Let X be a regular space and τ be a regular or an uncountable non-sequential cardinal number. Thefollowing are equivalent:

(1) k(X) � τ .(2) For every lower semi-continuous closed-valued mapping θ : X → Y into a complete metrizable space Y there

exists a lower semi-continuous selection φ : X → Y of θ such that k(clY φ(X)) � τ .(3) For every lower semi-continuous closed-valued mapping θ : X → Y into a complete metrizable space Y there

exists a single-valued selection g : X → Y such that k(clY g(X)) � τ .(4) For every lower semi-continuous mapping θ : X → Y into a discrete space Y there exists a single-valued selection

g : X → Y such that k(g(X)) � τ .(5) Every open cover of X contains a subcover of cardinality < τ .

Proof. (1) → (2) Assume that k(X) � τ . Consider the following cases:Case 1. τ = ℵ0.In this case the space X is compact and for every lower semi-continuous mapping θ : X → Y in a complete metric

space Y there exists a compact-valued lower semi-continuous mapping φ : X → Y and an upper semi-continuouscompact-valued mapping ψ : X → Y such that φ(x) ⊆ ψ(x) ⊆ clY θ(x) for every x ∈ X [8]. Since ψ is an uppersemi-continuous compact-valued mapping and X is a compact space ψ(X) is a compact subset of the space Y . In thiscase clY φ(X) ⊆ ψ(X).

Case 2. τ is a non-limit cardinal number.


In this case there exists a cardinal number λ such that λ+ = τ . It is obvious that k(Z) � τ if and only if l(Z) � λ

for every space Z. In this case the theorem is a corollary of Theorem 8.Case 3. τ is an uncountable limit non-sequential cardinal number.In this case in the proof of implication (1) → (2) of Theorem 8 we can assume that |An| < τ for every n ∈ N. Since

τ is a non-sequential uncountable cardinal number, there exists a cardinal number λ < τ such that |An| � λ for everyn ∈ N. Hence w(φ(X)) � λ < τ and k(φ(X)) = λ+ < τ .

One can establish implications (2) → (3) → (4) → (5) → (1) by analogy with the proofs of the similar implica-tions in Theorem 8. �Acknowledgements

We are grateful to the guest editors D. Repovš and P.V. Semenov, for giving us the possibility to publicly expressour deep respect and best wishes to E. Michael. Yet, our thanks goes to the referee for the constructive criticism.

References

[1] M.M. Choban, Many-valued mappings and Borel sets, I, Trudy Moskov. Mat. Obshch. 22 (1970) 229–250; Trans. Moscow Math. Soc. 22(1970) 258–280.

[2] M.M. Choban, Many-valued mappings and Borel sets, II, Trudy Moskov. Mat. Obshch. 23 (1970) 272–301; Trans. Moscow Math. Soc. 23(1970).

[3] M.M. Choban, General theorems on selections, Serdica. Bulg. Math. Publ. 4 (1978) 74–90.[4] M.M. Choban, V. Valov, On one theorem of Michael on selection, C. R. Acad. Bulg. Sci. 28 (7) (1975) 671–673.[5] R. Engelking, General Topology, PWN, Warszawa, 1977.[6] R. Engelking, Dimension Theory, PWN, Warszawa, 1978.[7] E. Michael, Continuous selections, I, Ann. Math. (2) 63 (2) (1956) 361–382.[8] E. Michael, A theorem on semi-continuous set-valued functions, Duke Math. J. 26 (4) (1959) 647–656.[9] E. Michael, Another note on paracompact spaces, Proc. Amer. Math. Soc. 8 (1957) 822–828.

[10] S.I. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980) 291–317.[11] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Acad. Publ., 1998.



On connections between some selection and approximation resultsfor multivalued maps ✩

Boris Gel’man ∗, Valeri Obukhovskii

Faculty of Mathematics, Voronezh State University, Universitetskaya pl. 1, 394006 Voronezh, Russia

Received 28 September 2006; received in revised form 13 February 2007; accepted 13 February 2007

This paper is dedicated to Professor Ernest Michael on the occasion of his jubilee.

Abstract

We study the general conditions on a family of values of a multivalued map under which the existence of single-valued approxi-mations may be derived from continuous selection results.© 2007 Elsevier B.V. All rights reserved.

MSC: 54C65; 54C60

Keywords: Multivalued map; Continuous selection; Single-valued approximation; Regular operation; Michael system

Introduction

Deep analogies and connections in constructions of continuous selections and ε-approximations of multivaluedmaps are well known beginning with classical results (see, e.g., [10,2]). There exists a number of collections ofsubsets (closed convex, decomposable, Rδ) with the property that lower semicontinuous multimaps taking the valuesin these families have continuous selections whereas corresponding upper semicontinuous multimaps admit single-valued approximations (see, e.g., [1,3,4,6,8,11]).

In the present paper we study the general conditions on a family of values of a multimap under which the existenceof single-valued approximations may be derived from continuous selection results. We develop and extend the resultsof the paper [5].

Let us mention also the works [12,14] and others (the complete bibliography may be found in [13]) where differentapproaches to the same problem were suggested.

The paper is organized in the following way. In the first section we introduce the notions of weakly regular andregular operations on subsets of a metric space and study their main properties. This notion is used in the next sectionto define an approximate family of sets. The general result on existence of a lower semicontinuous approximation foran upper semicontinuous multimap taking its values in an approximate family (Theorem 2.1) is proved. In the last

✩ The work is partially supported by the Russian FBR Grants No. 05-01-00100, 04-01-00081 and NATO Grant ICS.NR.CLG 981757.* Corresponding author.

E-mail addresses: [email protected] (B. Gel’man), [email protected] (V. Obukhovskii).


806 B. Gel’man, V. Obukhovskii / Topology and its Applications 155 (2008) 805–813

section we introduce the notions of a Michael system and a strong Michael system of subsets, present some examples,and give the general result on the existence of a single-valued ε-approximation for an upper semicontinuous multimapwith the values in a strong Michael system of subsets (Theorem 3.2). Some corollaries are presented.

1. Regular operations on sets in metric spaces

Let Y be a metric space. We will use the following notation. By P(Y ) we denote the collection of all nonemptysubsets of Y :

C(Y ) = {D ∈ P(Y ): D is closed

};K(Y) = {

D ∈ P(Y ): D is compact}.

If Y is a subset of a normed space, by V (Y ) [Cv(Y )] we denote the collection of all nonempty convex (respectively,closed convex) subsets of Y .

For D ∈ P(Y ) and ε > 0, the symbol Uε(D) denotes the ε-neighborhood of D.We will use the following notions.

Definition 1.1. A map λ :P(Y ) → P(Y ) is called the operation on subsets of a metric space Y if B ⊂ λ(B) for eachB ∈ P(Y ).

Definition 1.2. An operation λ :P(Y ) → P(Y ) is said to be weakly regular if the following conditions hold:

(i) if B,C ∈ P(Y ) and C ⊂ B then λ(C) ⊂ λ(B);(ii) for each ε > 0 and every natural number n there exists δ = δ(ε, n) > 0 such that for each set B ∈ P(Y ) and each

finite set K = {y1, y2, . . . , yn} ⊂ Uδ(B) we have λ(K) ⊂ Uε(λ(B));(iii) for each B ∈ P(Y ), point y ∈ λ(B), and every open neighborhood U of y there exists a finite subset N =

{z1, z2, . . . , zm} ⊂ B such that U ∩ λ(N) �= ∅.

Consider certain examples of weakly regular operations.

Example 1.1. Let cl :P(Y ) → P(Y ) be the operation of closure: cl(B) = B . It is easy to see that this operation isweakly regular.

Example 1.2. Let Y be a convex subset of a normed space E. A map co :P(Y ) → P(Y ) assigning to each set B ⊂ Y

its convex hull co(B) is weakly regular.

Example 1.3. Let I = [a, b] be an interval endowed with the Lebesgue measure; E a Banach space. By the symbolL1(I ;E) we denote a Banach space of (equivalence classes) of Bochner summable functions ϕ : I → E with the usualnorm

‖ϕ‖ =∫

I

∥∥ϕ(s)

∥∥

Eds.

For a measurable subset m ⊂ I , let κm : I → [0,1] denote its characteristic function

κm(t) ={

1, t ∈ m,

0, t /∈ m.

Let us recall the following notion (see, e.g., [1,3,4,8]).A set M ⊂ L1(I ;E) is called decomposable if for each ϕ,ψ ∈ M and every measurable subset m ⊂ I the function

κm · ϕ + κI\m · ψbelongs to M .

B. Gel’man, V. Obukhovskii / Topology and its Applications 155 (2008) 805–813 807

Denote by D(L1(I ;E)) the collection of all nonempty decomposable subsets of L1(I ;E). Consider a map

dec :P(L1(I ;E)

) → P(L1(I ;E)

)

assigning to each B ∈ P(L1(I ;E)) its decomposable hull

dec(B) =⋂{

C: B ⊂ C,C ∈ D(L1(I ;E)

)}.

It is easy to verify that this operation satisfies conditions (i)–(iii) of Definition 1.2.

Let us mention the following properties of weakly regular operations.

Proposition 1.1. Let (X,ρX), (Y,ρY ) be metric spaces; f :X → Y a homeomorphism satisfying the following condi-tion: there exist positive numbers c1 and c2 such that for each x, x′ ∈ X we have

c1ρX(x, x′) � ρY

(f (x), f (x′)

)� c2ρX(x, x′).

Let λ be a weakly regular operation on subsets of Y . Then the map λ′ :P(X) → P(X),

λ′(A) = f −1(λ(f (A)

)),

defines a weakly regular operation on subsets of X.

Proof. The validity of property (i) is evident.Let us verify property (ii). Fix ε > 0 and natural number n. For arbitrary subset C ⊂ X and any η > 0, take a finite

set K = {x1, x2, . . . , xn} ⊂ Uη(C) and let

K ′ = {f (x1), f (x2), . . . , f (xn)

}.

Denoting B = f (C), by assumption we have the following relation: K ′ ⊂ Uc2η(B). Since λ is a weakly regularoperation on subsets of Y we may find, for ε1 = c1ε, such δ1 > 0 that for each finite set S = {y1, y2, . . . , yn} ⊂ Uδ1(B)

we have λ(S) ⊂ Uε1(λ(B)). Let us assume that η < δ1c2

then λ(K ′) ⊂ Uε1(λ(f (C))). Now we obtain the followingrelations

λ′(K) = f −1(λ(K ′)) ⊂ f −1(Uε1

(λ(f (C)

))) ⊂ Uε

(f −1(λ

(f (C)

))) = Uε

(λ′(C)

)

proving property (ii).Property (iii) can be easily checked. �Let X,Y be metric spaces. Recall that a multivalued map (multimap) F :X → P(Y ) is said to be upper semicon-

tinuous (lower semicontinuous) provided the small preimage

F−1+ (V ) = {x ∈ X: F(x) ⊂ V

}

(respectively, the complete preimage

F−1− (V ) = {x ∈ X: F(x) ∩ V �= ∅}

)

of each open set V ⊂ Y is open in X (see, e.g., [1,3,4,6,8] for further details).For a given multimap F :X → P(Y ) and a weakly regular operation λ on subsets of Y , define the multimap

Fλ :X → P(Y ) by the rule

Fλ(x) = λ(F(x)

).

To study continuity properties of the multimap Fλ we will need the following assertion.

Lemma 1.1. A multimap F :X → P(Y ) is lower semicontinuous at a point x0 ∈ X iff for every compact set K ⊂ F(x0)

and each ε > 0 there exists δ = δ(x0,K, ε) > 0 such that ρX(x0, x) < δ implies K ⊂ Uε(F (x)).


Proof. (Necessity) Let F be lower semicontinuous at x0, K ⊂ F(x0) an arbitrary compact set, ε > 0 a given num-ber. Supposing the contrary, we will have sequences {xn} ⊂ X, xn → x0 and {yn} ⊂ K ⊂ F(x0) such that yn /∈Uε(F (xn)).

Since K is a compact set, we may assume, without loss of generality, that {yn} → y0 ∈ K . It is clear that y0 /∈Uε/2(F (xn)) for all sufficiently large n. For the open set V = Uε/2(y0), it is obvious that F(x0) ∩ V �= ∅ whereasthere does not exist an open neighborhood of a point x0 such that F(x) ∩ V �= ∅ for each x from this neighborhood,that gives the contradiction.

(Sufficiency) Consider an open set V ⊂ Y such that F(x0) ∩ V �= ∅. Take a point y0 ∈ F(x0) ∩ V . Let ε > 0 besuch that Uε(y0) ⊂ V . Since y0 is a compact subset of Y , by assumption there exists δ = δ(x0, y0, ε) > 0 such thatρ(x0, x) < δ implies y0 ⊂ Uε(F (x)). Whence F(x) ∩ V �= ∅ for each x ∈ Uδ(x0) yielding the semicontinuity of F

at x0. �Now we can verify the following property.

Theorem 1.1. Let λ :P(Y ) → P(Y ) be a weakly regular operation on subsets of a space Y . If a multimap F :X →P(Y ) is lower semicontinuous then the multimap Fλ is also lower semicontinuous.

Proof. For an arbitrary open set V ⊂ Y , let us demonstrate that the complete preimage (F λ)−1− (V ) is the open subsetof X. Assuming that this set is nonempty, for a given x0 ∈ (F λ)−1− (V ), take a point y0 ∈ λ(F (x0)) ∩ V . Let ε0 > 0 besuch that Uε0(y0) ⊂ V .

Since y0 ∈ λ(F (x0)), applying condition (iii) of Definition 1.2 we conclude that there exists a finite set N ={z1, z2, . . . , zm} ⊂ F(x0) such that λ(N) ∩ Uε0/2(y0) �= ∅.

By virtue of condition (ii) of the same definition there exists δ = δ(ε02 ,m) > 0 such that for each B ⊂ Y with

Uδ(B) ⊃ N we have λ(N) ⊂ Uε0/2(λ(B)).Since F is lower semicontinuous, there exists μ > 0 such that N ⊂ Uδ(F (x)) for all x, ρX(x, x0) < μ.Now to prove the assertion, it is sufficient to demonstrate that Uμ(x0) ⊂ (F λ)−1− (V ).In fact, for x ∈ Uμ(x0) we have N ⊂ Uδ(F (x)) and hence λ(N) ⊂ Uε0/2(F

λ(x)). This results(Uε0/2(y0) ∩ Uε0/2

(Fλ(x)

)) ⊃ (Uε0/2(y0) ∩ λ(N)

) �= ∅,

i.e., y0 ∈ Uε0(Fλ(x)) or, equivalently, Fλ(x) ∩ Uε0(y0) �= ∅ and therefore Fλ(x) ∩ V �= ∅. �

Now consider the following stronger class of operations.

Definition 1.3. Operation λ :P(Y ) → P(Y ) is called regular if it satisfies conditions (i) and (iii) of Definition 1.2 andcondition

(ii′) for every ε > 0 there exists δ = δ(ε) > 0 such that for each sets B ⊂ Y and K ⊂ Uδ(B) the following relationholds: λ(K) ⊂ Uε(λ(B)).

It is clear that each regular operation is weakly regular, but the inverse is not true in general. In fact, it is readilyseen that the operations from Examples 1.1 and 1.2 are regular whereas the operation of Example 1.3 is not of thistype.

Let us mention some properties of regular operations.

Proposition 1.2. Let λ1, λ2 :P(Y ) → P(Y ) be regular operations; then their composition λ = λ0 ◦λ1 :P(Y ) → P(Y ),

λ(B) = λ2(λ1(B)

)

is a regular operation on subsets of Y .

Proof. The validity of properties (i) and (iii) is evident. Let us verify property (ii′). For a given B ⊂ Y and ε > 0 thereexists δ1 > 0 such that for each set S ⊂ Uδ1(λ1(B)) we have λ2(S) ⊂ Uε(λ2(λ1(B)). From the other side, there exists


δ > 0 such that for each set K ⊂ Uδ(B) the relation λ1(K) ⊂ Uδ1(λ1(B)) is true. Then, if K ⊂ Uδ(B), we have

λ2(λ1(K)

) ⊂ Uε

(λ2

(λ1(B)

))

proving property (ii′). �Notice that from this assertion it follows that the operation of a closed convex hull λ(B) = co(B) is regular.

Proposition 1.3. Let λ :P(Y ) → P(Y ) be a regular operation. If λ transfers each compact set A ⊂ Y to the compactset λ(A), then for each upper semicontinuous multimap F :X → K(Y), the multimap Fλ :X → K(Y) is also uppersemicontinuous.

Proof. Let V ⊂ Y be any open set. Let us show that its small preimage (F λ)−1+ (V ) is the open subset of X. In fact,assuming that (F λ)−1+ (V ) is nonempty, take any point x0 ∈ (F λ)−1+ (V ), then Fλ(x0) ⊂ V . Since Fλ(x0) is a compactset, there exists a number η > 0 such that Uη(F

λ(x0)) ⊂ V . Applying the upper semicontinuity of F and property (ii′)of a regular operation we conclude that there exist numbers ε > 0 and δ = δ(η) > 0 such that for each point x ∈ X

with ρX(x, x0) < ε we have: F(x) ⊂ Uδ(F (x0)) and λ(Uδ(F (x0))) ⊂ Uη(λ(F (x0))) ⊂ V . Therefore

Fλ(x) = λ(F(x)

) ⊂ Uη

(λ(F(x0)

)) ⊂ V

provided ρX(x, x0) < ε proving the assertion. �2. Approximate families of sets

Let Y be a metric space; λ :P(Y ) → P(Y ) an operation on subsets of Y . A set B ⊂ Y is called invariant withrespect to the operation λ if λ(B) = B . The collection of all invariant subsets with respect to λ is called the kernel ofoperation λ.

Definition 2.1. Let an operation λ be regular and λ(B) belong to the kernel of λ for each B ∈ P(Y ). Then the kernelof λ will be called an approximate family with respect to the operation λ and it will be denoted by Aλ(Y ).

It is clear that if Aλ(Y ) is the approximate family with respect to λ then the range of λ coincides with Aλ(Y ).Notice that the collection C(Y ) is the approximate family with respect to the operation of closure λ(B) = B . If

Y is a closed convex subset of a normed space, then the collections V (Y ) and Cv(Y ) are approximate families withrespect to the operations of a convex (respectively, closed convex) hull: λ(B) = co(B) [λ(B) = co(B)].

Let us mention the following property of approximate families.

Proposition 2.1. Let Aλ(Y ) be an approximate family. Then for each ε > 0 there exists δ > 0 such that C ∈ Aλ(Y )

and B ⊂ Uδ(C) implies λ(B) ⊂ Uε(C).

Proof. Let δ > 0 corresponds to a given ε in accordance with property (ii′). Then for given B ⊂ Uδ(C), C ∈ Aλ(Y )

we have

λ(B) ⊂ λ(Uδ(C)

) ⊂ Uε

(λ(C)

) = Uε(C). �Approximate families play an important role in constructions of approximations of multivalued maps.Let (X,ρX), (Y,ρY ) be metric spaces; the metric ρ in the Cartesian product X ×Y may be defined in the following

way:

ρ((x, y), (x′, y′)

) = max{ρX(x, x′), ρY (y, y′)

}.

Definition 2.2. For a given ε > 0, a lower semicontinuous multimap Fε :X → P(Y ) is called a lower semicontinuousε-approximation of a multimap F :X → P(Y ) provided the following conditions hold:


(i) F(x) ⊂ Fε(x) for each x ∈ X;(ii) Γ (Fε) ⊂ Uε(Γ (F )), where Γ denotes the graph of corresponding multimap.

Let us recall also that a continuous map f :X → Y is said to be a single-valued ε-approximation of a multimapF :X → P(Y ) provided Γ (f ) ⊂ Uε(Γ (F )) (see, for example, [1–6,8,9]).

The following general statement on the existence of a lower semicontinuous approximation holds true.

Theorem 2.1. Let Aλ(Y ) be an approximate family of subsets of a metric space Y corresponding to an operation λ.If F :X → Aλ(Y ) is an upper semicontinuous multimap then for each ε > 0 there exists a lower semicontinuousε-approximation Fε :X → Aλ(Y ) such that Fε(X) ⊂ λ(F (X)).

To prove this theorem we will need the following additional notions.Let X be a metric space; σ = {Uα}α∈J a cover of X consisting of open sets. Denote by St(x) the star of a point

x ∈ X with respect to the cover σ , i.e.,

St(x) =⋃

α∈Λ(x)

Uα, where Λ(x) = {α ∈ J | x ∈ Uα}.

For a given multimap F :X → P(Y ) and a cover σ of X we may define a multimap Fσ :X → P(Y ) by

Fσ (x) = F(St(x)

).

It is easy to verify the following.

Lemma 2.1. For each multimap F the multimap Fσ is lower semicontinuous.

Proof of Theorem 2.1. For a given ε > 0 let us take a number δ = δ(ε) > 0 corresponding to ε by virtue ofProposition 2.1. Since F is upper semicontinuous, for each point x ∈ X there exists η(x) > 0 such that for everypoint x′ ∈ Uη(x)(x) we have F(x′) ⊂ Uδ(F (x)). We may assume without loss of generality, that 0 < η(x) < ε. Setμ(x) = 1

4η(x). Consider the cover {Uμ(x)(x)}x∈X and extract a locally finite subcover σ = {Uμj(xj )}j∈J . Now define

a multivalued map Fε by the following way:

Fε(x) = λ(Fσ (x)

).

Let us demonstrate that the multimap Fε is the desirable one.In fact, applying Lemma 2.1 and Theorem 1.1 we conclude that this multimap is lower semicontinuous. By using

Proposition 2.1 it can easily be checked that F(x) ⊂ Fσ (x) ⊂ Fε(x) for all x ∈ X.Let us demonstrate that the multimap Fε satisfies condition (ii) of Definition 2.2. For an arbitrary point x ∈ X

take its star St(x) = ⋃kj=1 Uμj

(xj ). Let μs = max1�j�k μj , then x ∈ Uμs (xj ) for each j = 1,2, . . . , k. Hence xj ∈U2μs (xs) for each j = 1,2, . . . , k. Therefore St(x) ⊂ U3μs (xs) ⊂ Uη(xs)(xs). From the definition of the set Uη(xs)(xs)

it follows that

F(x′) ⊂ Uδ

(F(xs)

)

for each x′ ∈ St(x). So we have Fσ (x) = F(St(x)) ⊂ Uδ(F (xs)). Applying property (ii′) of regular operation weobtain

Fε(x) = λ(Fσ (x)

) ⊂ Uε

(F(xs)

).

Since ρ(x, xs) < ηs < ε the following relation holds:

Γ (Fε) ⊂ Uε

(Γ (F)

).

The fulfillment of the condition Fε(X) ⊂ λ(F (X)) follows from the definition of the multimap Fε . �Remark 2.1. The validity of Theorem 2.1 for weakly regular operations is an open question.


3. Michael systems

Now we introduce the following concept.

Definition 3.1. A family M(Y) of subsets of a metric space Y is called a Michael system if for every metric space X,lower semicontinuous multimap F :X → M(Y), a closed subset A ⊂ X, and a continuous selection f :A → Y of therestriction F |A there exists a continuous selection g :X → Y of the multimap F such that g|A = f .

Let us consider some examples of Michael systems.From the classical Michael theorem [10] it follows that the collection Cv(Y ), where Y is a closed subset of a Banach

space is a Michael system. Moreover, let Z be a metric space homeomorphic to Y and g :Y → Z a homeomorphism.It is easy to see that the family of sets

Mg(Z) = {g(A): A ∈ Cv(Y )

}

is a Michael system.In the work [7] the authors studied the family G(Y) of subsets of a metric space Y described by the following set

of axioms:

(G1) Y ∈ G(Y) and {y} ∈ G(Y) for each point y ∈ Y ;(G2) If {Ai}i∈I ⊂ G(Y) then

⋂i∈I Ai ∈ G(Y);

(G3) For each finite set y1, y2, . . . , yk ∈ Y the set

μ(y1, y2, . . . , yk) =⋂{

A: A ∈ G(Y), y1, y2, . . . , yk ∈ A}

is of infinite connectivity;(G4) For each ε > 0 there exists δ > 0 such that for every A ∈ G(Y) and a finite set z1, z2, . . . , zm ∈ Uδ(A) we have

μ(z1, z2, . . . , zm) ⊂ Uε(A);(G5) For each A ∈ G(Y), y ∈ Y , and r > 0 the following relation holds:

A ∩ Br [y] ∈ G(Y).

If a family G(Y) satisfies conditions (G1)–(G5), then we say that this family is of G-type.

Now one of the results of [7] may be formulated in our terms in the following way.

Theorem 3.1. If a space Y is complete and a family G(Y) is of G-type then this family is a Michael system.

Other examples of Michael systems also may be found in the same paper.Describe the connection between families of that type and regular operations on subsets.

Proposition 3.1. Suppose that a family G(Y) satisfies axioms (G1), (G2), (G4) and, additionally,

(G6) the set⋃{

μ(y1, y2, . . . , yk): y1, y2, . . . , yk ∈ A}

is dense in⋂{

B: B ∈ G(Y),A ⊂ B}.

Then there exists a weakly regular operation μ on subsets of Y such that its kernel coincides with G(Y) and μ◦ μ = μ.

Proof. Define the map μ :P(Y ) → P(Y ) by

μ(A) =⋂{

B: B ∈ G(y),A ⊂ B}.


Let us verify that this map is a weakly regular operation. Using axiom (G1) we get A ⊂ μ(A), i.e., the map μ is theoperation. Further, the fulfillment of property (i) of Definition 1.2 for μ is evident. Property (ii) follows from (G4)whereas property (iii) is the consequence of (G6). Axiom (G2) implies that G(Y) is the kernel of the operation μ andμ ◦ μ = μ. �

If we slightly change axiom (G4) we will achieve that the family G(Y) is the kernel of a regular operation. In fact,it is easy to verify the following statement.

Proposition 3.2. Let a family G(Y) satisfy axioms (G1), (G2), (G6) and

(G4′) For each ε > 0 there exists δ > 0 such that for every A ⊂ Y and K ⊂ Uδ(A) we have

μ(K) ⊂ Uε

(μ(A)

),

where μ is defined as above.

Then μ is a regular operation on P(Y ) and G(Y) is the approximate family with respect to μ.

Now let us introduce the following notion.

Definition 3.2. If a Michael system M(Y) is an approximate family with respect to a regular operation λ then it iscalled a strong Michael system and is denoted by AMλ(Y ).

This concept is connected with the problem of existence of single-valued approximations for multimaps. In fact,the following statement holds true.

Theorem 3.2. If a multimap F :X → AMλ(Y ) is upper semicontinuous then, for each ε > 0, it has a single-valuedε-approximation fε :X → Y such that fε(X) ⊂ λ(F (X)).

Proof. Prom Theorem 2.1 we know that there exists a lower semicontinuous ε-approximation Fε :X → AMλ(Y ) suchthat Fε(X) ⊂ λ(F (X)). But since AMλ(Y ) is a Michael system the multimap Fε has a continuous selection fε whichis the desired ε-approximation of F . �

As the consequence we get the following known result on existence of single-valued approximations (cf. [2,9]).

Corollary 3.1. If X is a metric space, E a Banach space, and F :X → Cv(E) an upper semicontinuous multimapthen, for each ε > 0, there exists a single-valued ε-approximation fε :X → E of F such that fε(X) ⊂ co(F (X)).

The application of Proposition 3.2 and Theorems 3.1 and 3.2 yields the following assertion.

Corollary 3.2. Let X and Y be metric spaces; a family of subsets G(Y) satisfy conditions (G1)–(G3), (G4′), (G5),and (G6). Then, for every upper semicontinuous multimap F :X → G(Y) and ε > 0, there exists a single-valuedε-approximation fε :X → Y of F such that fε(X) ⊂ μ(F(X)).

Corollary 3.3. Let Y be a metric space, A ⊂ Y a closed subset, and AMλ(Y ) a strong Michael system. If a multimapF :X → AMλ(Y ) is upper semicontinuous and a map f :A → Y is a continuous selection of the restriction F |A then,for every ε > 0, there exists a single-valued ε-approximation fε :X → Y of F such that fε(x) = f (x) for all x ∈ A

and fε(X) ⊂ λ(F (X)).

Proof. By virtue of Theorem 2.1 there exists a lower semicontinuous ε-approximation Fε :X → AMλ(Y ). SinceAMλ(Y ) is a strong Michael system we may choose a continuous selection of Fε being a continuous extension of f .It is clear that this selection is the desirable single-valued ε-approximation. �


References

[1] Y.G. Borisovich, B.D. Gel’man, A.D. Myshkis, V.V. Obukhovskii, Introduction to Theory of Multivalued Maps and Differential Inclusions,KomKniga, Moscow, 2005 (in Russian).

[2] A. Cellina, Approximation of set-valued functions and fixed-point theorems, Ann. Mat. Pura Appl. (4) 82 (1969) 17–24.[3] K. Deimling, Multivalued Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 1, Walter de Gruyter, Berlin–

New York, 1992.[4] A. Fryszkowski, Fixed Point Theory for Decomposable Sets, Topological Fixed Point Theory and its Applications, vol. 2, Kluwer Academic

Publishers, Dordrecht, 2004.[5] B.D. Gel’man, H.R. Al-Hashemi, On approximations of multivalued maps, Vestnik Voronezh. Gos. Univ. Ser. Fiz., Matem. 2 (2003) 136–143

(in Russian).[6] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, vol. 495, Kluwer Academic

Publishers, Dordrecht, 1999.[7] L. Górniewicz, S.A. Marano, M. Slosarski, Fixed points of contractive multivalued maps, Proc. Amer. Math. Soc. 124 (9) (1996) 2675–2683.[8] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter

Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin–New York, 2001.[9] A. Lasota, Z. Opial, An approximation theorem for multi-valued mappings, Podstawy Sterowania 1 (1) (1971) 71–75.

[10] E. Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956) 361–382.[11] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, Mathematics and its Applications, vol. 455, Kluwer Academic

Publishers, Dordrecht, 1998.[12] D. Repovš, P.V. Semenov, On relative approximation theorems, Houston J. Math. 28 (3) (2002) 497–509.[13] D. Repovš, P.S. Semenov, Continuous selections of multivalued mappings, in: Recent Progress in General Topology, vol. II, North-Holland,

Amsterdam, 2002, pp. 423–461.[14] E.V. Shchepin, N.B. Brodskii, Selections of filtered multivalued mappings, Trudy Mat. Inst. Steklov. Otobrazh. i Razmer. 212 (1996) 220–240

(in Russian); transl. in: Proc. Steklov Inst. Math. 212 (1) (1996) 209–229.



Selections and topological convexity ✩

Valentin Gutev

School of Mathematical Sciences, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa

Received 22 August 2006; received in revised form 21 January 2007; accepted 21 January 2007

Abstract

A simple natural proof of van de Vel’s selection theorem for topological convex structures is given. The technique developed toachieve this proof allows to give also a direct simple proof of the classical Michael’s selection theorem in Fréchet spaces, and theHorvath’s selection theorem in metric l.c.-spaces.© 2007 Elsevier B.V. All rights reserved.

MSC: primary 54C60, 54C65, 52A01, 54F35; secondary 54B20, 54D15

Keywords: Set-valued mapping; Selection; Homotopically trivial set; c-system; Convex structure; Hyperspace topology

1. Introduction

As far as convex structures are concerned, our notation and terminology follow that one of [19,20]. A family C ofsubsets of a set Y is called a convexity on Y if

conv 1: ∅, Y ∈ C.conv 2:

⋂D ∈ C for every nonempty D ⊂ C.

conv 3:⋃

D ∈ C for every nonempty D ⊂ C which is totally ordered by inclusion.

The pair (Y,C) is called a convex structure, while the members of C are called convex sets. According to theaxioms of a convex structure, to any subset A ⊂ Y one can associate a convex set

co(A) =⋂

{C ∈ C: A ⊂ C},which is called the convex hull of A. For a finite F ⊂ Y , the convex hull co(F ) is called a polytope.

A half-space H of Y is a convex set H ∈ C such that Y \ H is also convex, i.e. Y \ H ∈ C. There is a list ofseparation axioms for convex structures, comparable with the separation axioms in topology. Two of them are thefollowing:

S1: All singletons are convex, i.e. {y} ∈ C for every y ∈ Y .

✩ This work is based upon research supported by the NRF of South Africa.E-mail address: [email protected].


V. Gutev / Topology and its Applications 155 (2008) 814–823 815

S4: Disjoint convex sets can be extended to complementary half-spaces, i.e. if C,D ∈ C are disjoint, then there existsa half-space H ∈ C such that C ⊂ H and D ⊂ Y \ H .

The axiom S1 will be assumed in all considerations.A topological convex structure (Y,T ,C) is a set Y equipped with both a convexity C and a topology T such that

all polytopes are closed. If, in addition, the closure C of each convex set C ∈ C is convex, then the structure is calledclosure stable.

In what follows, for a subset C ⊂ Y and a cover U of Y , we use st(C,U) to denote the star of C with respect toU. A (covering) uniformity μ on Y is called compatible with a convex structure (Y,C) if all polytopes are closed inthe uniform topology, and for each uniform cover U ∈ μ there is a uniform cover V ∈ μ such that

co(st(C,V)

) ⊂ st(C,U), whenever C ∈ C. (1.1)

In this case, the triple (Y,μ,C) is called a uniform convex structure. A topological convex structure is uniformizable ifthe topology is generated by a uniformity which is compatible with the convexity. Finally, let us recall that a uniformconvex structure (Y,μ,C) is called metrizable if the uniformity μ is generated by a metric d . In this case, the metricd is said to be compatible.

Let (Y,C) be a metrizable convex structure, and let d be a compatible metric. For a nonempty subset A ⊂ Y

(a point y ∈ Y ) and ε > 0, we let Bdε (A) (respectively, Bd

ε (y)) to be the open ε-neighborhood of A (respectively, y).Then, by definition, for every ε > 0 there exists a δ > 0 such that co(Bd

δ (C)) ⊂ Bdε (C) for each nonempty convex

set C ∈ C. According to [19, Theorem 2.6], this implies that for each C ∈ C there exists a convex open set O , withBd

δ (C) ⊂ O ⊂ Bdε (C). However, in general, the open balls themselves need not be convex (not even the balls centered

at one point).In what follows, we will use Φ : X � Y to denote that Φ is a set-valued mapping, i.e. a mapping from X into the

nonempty subsets of Y . Recall that Φ is lower semi-continuous, or l.s.c., if the set

Φ−1(U) = {x ∈ X: Φ(x) ∩ U �= ∅}

is open in X for every open U ⊂ Y . Finally, recall that Φ : X � Y (respectively, a single-valued map f : X → Y ) is aselection for Ψ : X � Y if Φ(x) ⊂ Ψ (x) (respectively, f (x) ∈ Ψ (x)) for every x ∈ X.

The following selection theorem was proved by van de Vel in [19] (see, also, [20]).

Theorem 1.1. (See [19].) Let X be a paracompact space, Y be a metrizable S4 convex structure with compact poly-topes and with connected convex sets, and let d be a compatible metric on Y . Then, each l.s.c. mapping Φ : X � Y

with convex and d-complete values admits a single-valued continuous selection.

It should be remarked that two well-known selection theorems are plainly included in Theorem 1.1: the classicalMichael’s selection theorem in Banach and Fréchet vector spaces [12–14], and Nadler’s result on selecting fromsubcontinua of a compact metric tree [18]. On the other hand, we refer the interested reader to [19,20] for a variety ofother applications of Theorem 1.1.

The purpose of the present paper is to provide a simple geometrical proof of Theorem 1.1. The preparation for thisproof is done in Sections 2 and 3. Briefly, Section 2 deals with two simple results about selections for c-systems. Itshould be mentioned that the concept of a c-system is very closed to the one of a c-structure introduced by Horvath[9,10]. The difference between them is very small, but it will be crucial to achieve the promised proof of Theorem 1.1.The second element of our approach is summarized in Section 3, and presents the van de Vel’s technique [19,20]of “small selections” to construct single-valued continuous selections of l.s.c. mappings. The proof of Theorem 1.1will be finally accomplished in Section 4, where are demonstrated also some other applications about selections andtopological convexity.

The technique of “small selections” is very useful to give another direct and simple proof of a known selectiontheorem—the classical Michael’s selection theorem in Fréchet spaces [14]. This is done in Section 5. In fact, ourarguments are almost the same as the original Michael’s arguments in [13] for the case of Banach spaces, but we donot rely on Cauchy sequences of single-valued continuous approximations. Consequently, we do not rely on boundedsets of topological vector spaces, hence we do not require these spaces to be normable.

816 V. Gutev / Topology and its Applications 155 (2008) 814–823

Finally, based on the same idea, we give also a simple proof of the Horvath’s selection theorem in complete metricl.c.-spaces [10], see Section 6.

In conclusion, the author would like to express his best gratitude to Professor Vesko Valov for several valuableremarks concerning the proof of Lemma 4.1.

2. Selections and homotopically trivial spaces

By a simplicial complex we mean a collection Σ of nonempty finite subsets of a set S such that ∅ �= τ ⊂ σ ∈ Σ

implies τ ∈ Σ . The elements of the set V(Σ) = ⋃Σ are usually called vertices of Σ , while any element of Σ is

called a simplex of Σ . Whenever k < ω, we let Σk to be the k-skeleton of Σ , i.e. the simplicial complex Σk ={σ ∈ Σ : Card(σ ) � k + 1}.

The set of all nonempty finite subsets of S is a simplicial complex. In the sequel, we will denote it by ΣS . An-other natural example of a simplicial complex is the nerve N (U) of a point-finite cover U of a set Z, which is thesubcomplex of ΣU defined by

N (U) ={σ ⊂ U:

⋂σ �= ∅

}=

{σ ∈ ΣU:

⋂σ �= ∅

}.

Motivated by this, we will consider the nerve of an arbitrary cover U of Z by letting that N (U) = {σ ∈ ΣU:⋂

σ �= ∅}.The vertices of a simplicial complex Σ can be identified as a linearly independent subset of some linear normed

space. Then, to any simplex σ ∈ Σ we may associate the geometric simplex |σ | which is the convex hull of σ . Thus,Card(σ ) = k + 1 if and only if |σ | is a k-dimensional simplex. Finally, let |Σ | = ⋃{|σ |: σ ∈ Σ}, which is usuallycalled the geometric realization of Σ . As a topological space, we will consider |Σ | endowed with the Whiteheadtopology. Recall that in this topology a subset U ⊂ |Σ | is open if and only if U ∩ |σ | is open in |σ | for every σ ∈ Σ .

Let Sk be the k-dimensional sphere, and B

k—the k-dimensional ball. Recall that a space Z is homotopically trivialif, for every k < ω, every continuous map f : S

k → Z can be extended to a continuous map g : Bk+1 → Z. Also, let

us recall that a c-structure on a space Z [9,10] is a set-valued mapping : ΣZ � Z such that, for every σ, τ ∈ ΣZ ,

(i) (τ) ⊂ (σ ) provided τ ⊂ σ (i.e., is monotone),(ii) (σ ) is a homotopically trivial subset of Z.

The pair (Z,) is called a c-space, [9,10].It should be remarked that the original definition of a c-structure requires each (σ ), σ ∈ ΣZ , to be a contractible

subset of Z rather than homotopically trivial, but, as mentioned in [10], the known results for such structures remaintrue if that requirement is weakened to homotopical triviality.

In the present section, we are interested in partial c-structures. To this end, for a set S, we shall say that a set-valuedmapping : ΣS � Z is a c-system if is monotone, and each (σ ), σ ∈ ΣS , is a homotopically trivial subset of Z.Clearly, every c-structure : ΣZ � Z is a c-system.

While the difference between c-systems and c-structures may look quite small, we will see in Section 4 that itis crucial to achieve an easy geometrical proof of Theorem 1.1. Here, we are interested in the following selectionproperties of c-systems.

Lemma 2.1. Let Z be a space, S be a set, and let : ΣS � Z be a c-system. Then, there exists a continuous mapv : |ΣS | → Z such that v(|σ |) ⊂ (σ ) for every σ ∈ ΣS .

Proof. For each n < ω, we will define a continuous map vn : |ΣnS | → Z such that

vn+1 is an extension of vn, (2.1)

vn(|σ |) ⊂ (σ ), whenever σ ∈ ΣnS . (2.2)

This can be done as follows. First, define an arbitrary map v0 : S = |Σ0S | → Z by v0(s) ∈ ({s}), s ∈ S. Next, proceed

by induction. Namely, suppose that vn is as in (2.2), and take a simplex σ ∈ Σn+1. If σ ∈ Σn, then let v(n+1,σ ) =
S S


vn�|σ |. If σ /∈ ΣnS , then |σ |∩ |Σn

S | is homeomorphic to the n-dimensional sphere, |σ | is homeomorphic to the (n+1)-dimensional ball, while, by (2.2),

vn

(|σ | ∩ |ΣnS |) ⊂

⋃{(τ): τ ⊂ σ and τ ∈ Σn

S

} ⊂ (σ ).

Hence, in this case, there exists a continuous extension v(n+1,σ ) : |σ | → (σ ) of vn � (|σ | ∩ |ΣnS |) because (σ ) is

homotopically trivial. We complete the construction of vn+1 by letting that vn+1 � |σ | = v(n+1,σ ), σ ∈ Σn+1S . In fact,

this also completes the proof because we can now take v � |ΣnS | = vn for every n < ω. �

Motivated by a result in [4], we shall say that a mapping Φ : X � S from a space X into the subsets of a set S is aBrowder mapping if the set

Φ−1(s) = {x ∈ X: s ∈ Φ(x)

}

is open in X for every point s ∈ S. Also, to any set-valued mapping Φ : X � S we will associate another one ΣΦ :X � ΣS defined by

ΣΦ(x) = ΣΦ(x) = {σ ⊂ Φ(x): σ ∈ ΣS

}, x ∈ X.

Note that ΣΦ(x) is a simplicial subcomplex of ΣS for every x ∈ X. Thus, we can also associate the mapping |ΣΦ | :X � |ΣS |, which is defined by

|ΣΦ |(x) = ∣∣ΣΦ(x)∣∣, x ∈ X.

Lemma 2.2. Let X be a paracompact space, S be a set, and let Φ : X � S be a Browder mapping. Then, |ΣΦ | has acontinuous selection g : X → |ΣS |.

Proof. Take an arbitrary selection s : X → S for Φ . Next, for every x ∈ X, let

Wx = Φ−1(s(x)) = {

y ∈ X: s(x) ∈ Φ(y)},

which is a neighborhood of x because Φ is a Browder mapping. Since X is paracompact, there exists a locally finiteopen cover U of X and a map p : U → X such that U ⊂ Wp(U) for every U ∈ U. Next, define a map π : N (U) → ΣS

by π(σ) = s(p(σ )), σ ∈ N (U). Then, whenever σ ∈ N (U),

x ∈⋂

σ implies π(σ) ⊂ Φ(x), (2.3)

because x ∈ ⋂σ ⊂ ⋂{Wp(U): U ∈ σ }. Finally, let {gU : U ∈ U} be a partition of unity on X, index-subordinated to

the cover U of X (i.e., g−1U ((0,1]) ⊂ U , U ∈ U). The canonical map g : X → |ΣS |, defined by

g(x) =∑{

gU(x) · π(U): U ∈ U}, x ∈ X,

is as required. Indeed, take a point x ∈ X, and let σx = {U ∈ U: x ∈ U}. Then, x ∈ ⋂σx and, by (2.3), π(σx) ⊂ Φ(x).

Therefore, g(x) ∈ |π(σx)| ⊂ |ΣΦ |(x). �Whenever Φ : X � S and ϕ : S � Z, we let ϕ ◦ Φ : X � Z to be the composition of these mappings, i.e.

ϕ ◦ Φ(x) =⋃{

ϕ(y): y ∈ Φ(x)}, x ∈ X.

Thus, to any c-system : ΣS � Z, and mapping Φ : X � S one can associate the set-valued mapping co [Φ] : X �Z (suggesting the “-convex hull” of Φ) by letting co [Φ] = ◦ ΣΦ .

The following immediate consequence of Lemmas 2.1 and 2.2 is a generalization of [4, Theorem 1]. For a finitedimensional-version of this result, see [8, Theorem 3.1].

Corollary 2.3. Let X be a paracompact space, Z be a space, : ΣS � Z be a c-system for some set S, and letΦ : X � S be a Browder mapping. Then, co [Φ] has a single-valued continuous selection.

Proof. By Lemma 2.1, there exists a continuous map v : |ΣS | → Z such that v(|σ |) ⊂ (σ ) for every σ ∈ ΣS . On theother hand, by Lemma 2.2, |ΣΦ | has a continuous selection g : X → |ΣS |. Then, f = v ◦ g : X → Z is a continuousselection for co [Φ]. �


3. Small selections

In the present section we summarize the approach of [19] to construct single-valued continuous selections forl.s.c. mapping. To this end, suppose that X and Y are spaces, and {ϕn: n < ω} is a sequence of set-valued mappingsϕn : X � Y , n < ω. We shall say that {ϕn: n < ω} is decreasing if ϕn+1(x) ⊂ ϕn(x) for every x ∈ X and n < ω.Also, if Y is metrizable and d is a metric on Y , compatible with the topology of Y , then we will use diamd(B) todenote the diameter of a subset B ⊂ Y with respect to d . In this case, for a set-valued mapping ϕ : X � Y we letdiamd(ϕ) = sup{diamd(ϕ(x)): x ∈ X}.

Finally, to any mapping ϕ : X � Y we will associate another one ϕ : X � Y defined by ϕ(x) = ϕ(x), x ∈ X.According to [13, Proposition 2.3], ϕ is l.s.c. if and only if ϕ is l.s.c.

Proposition 3.1. (See [19].) Let X be a space, (Y, d) be a complete metric space, and let {ϕn: n < ω} be a decreasingsequence of l.s.c. mappings ϕn : X � Y such that limn→∞ diamd(ϕn) = 0. Then, each ϕ(x) = ⋂{ϕn(x): n < ω},x ∈ X, is a singleton and the map f : X → Y , defined by {f (x)} = ϕ(x), x ∈ X, is continuous.

Proof. According to the Cantor Theorem (see [6]), ϕ(x) �= ∅ for every x ∈ X. Hence, each ϕ(x), x ∈ X, is a singletonbecause limn→∞ diamd(ϕn) = 0. To show that f is continuous, take a point x0 ∈ X and an ε > 0. Then, there existsan m < ω with diamd(ϕm) < ε/2. Set U = ϕ−1

m (Bdε/2(f (x0))) and observe that it is a neighborhood of x0 because

ϕm is l.s.c. and f (x0) ∈ ϕm(x0). Since f is a selection for ϕm and diamd(ϕm) = diamd(ϕm) < ε/2, this implies thatf (U) ⊂ Bd

ε (f (x0)) which completes the proof. �4. Small selections and topological convexity

Suppose that Σ is a simplicial complex. For every x ∈ |Σ | we consider the carrier of x, defined by

car(x) =⋂{

σ ∈ Σ : x ∈ |σ |}.It is well known that car(x) is the smallest simplex of Σ that contains x in its geometrical realization (see, for instance,[15, Corollary 2.1.17]). Now, for every simplex σ ∈ Σ and vertex s ∈ σ , let α(s,σ ) : |σ | → [0,1] be the correspondingaffine coordinate function of s in σ (see, for instance, [15, Theorem 2.1.8]). Take another simplex τ ∈ Σ . If s /∈ τ

and σ ∩ τ �= ∅, then α(s,σ )(x) = 0 for every x ∈ |σ ∩ τ |. If s ∈ τ , then α(s,σ ) � |σ ∩ τ | = α(s,τ) � |σ ∩ τ |. Hence, wecan extend α(s,σ ) to a continuous function αs : |Σ | → [0,1] by letting for σ ∈ Σ that αs � |σ | = α(s,σ ) if s ∈ σ andαs � |σ | ≡ 0 if s /∈ σ . Thus, for every s ∈ V(Σ), we can consider the affine coordinate function αs : |Σ | → [0,1] of s

in Σ . In these terms, for every x ∈ |Σ | we have that

car(x) = {s ∈ V(Σ): αs(x) �= 0

}. (4.1)

Recall that a cover O of a set Y is a star-refinement of another cover V of Y if {st(O,O): O ∈ O} is a refinementof V . Finally, to any cover V of Y , we will associate the mapping ΩV : N (V) � Y defined by

ΩV(σ ) =⋂

σ, σ ∈ N (V).

Lemma 4.1. Let X be a paracompact space, (Y,C) be a closure stable and S4 topological convex structure withcompact polytopes and with connected convex sets, ϕ : X � Y be an l.s.c. convex-valued mapping, and let V be anopen cover of Y . Also, let O be an open cover of Y which is a star-refinement of V , and let U be an open convexcover of Y which is a refinement of O and co(st(C,U)) ⊂ st(C,O) for every convex set C ∈ C. Then, there exists amap π : X → N (V) such that, for every x ∈ X,

(a) ϕ(x) ∩ ΩV(π(x)) �= ∅,(b) π(z) ⊂ π(x) for every z ∈ W and some neighborhood W of x.

Proof. In order to apply Corollary 2.3, we take S = U and Z = |N (U)|, and we are going to define a c-system : ΣU � |N (U)|. Namely, let p : U → Y be a map such that p(U) ∈ U for every U ∈ U, and let ΩU : N (U) � Y


be the mapping associated to the nerve N (U). Next, to any simplex σ ∈ ΣU we associate a subcomplex Γσ ⊂ N (U)

by letting that

Γσ = {τ ∈ N (U): ΩU(τ ) ∩ co(p(σ )) �= ∅}

.

Finally, set (σ ) = |Γσ |, σ ∈ ΣU. According to [19, Theorem 4.1], each (σ ), σ ∈ ΣU, is contractible, and clearly

is monotone. Thus, we get a c-system : ΣU � |N (U)|. Now, define a mapping Φ : X � U by letting that

Φ(x) = {U ∈ U: ϕ(x) ∩ U �= ∅}

, x ∈ X,

which is clearly a Browder mapping because ϕ is l.s.c. Hence, by Corollary 2.3, co [Φ] = ◦ ΣΦ has a continuousselection f : X → |N (U)|.

In order to use the selection f , we define another mapping Ψ : X � N (V) by letting that

Ψ (x) = {τ ∈ N (V): ϕ(x) ∩ ΩV(τ ) �= ∅}

, x ∈ X, (4.2)

where ΩV : N (V) � Y is the mapping associated to the nerve N (V).Since U is a refinement of O, there exists a map r : U → O with U ⊂ r(U), U ∈ U. On the other hand, O is a

star-refinement of V . Hence, there also exists a map s : O → V such that st(O,O) ⊂ s(O), O ∈ O. Thus, in fact, weget that

⋃δ ⊂

⋂s(δ) for every δ ∈ N (O). (4.3)

Finally, let � : N (U) → N (V) be the map generated by s ◦ r : U → V , and let us observe that

�(Γσ ) ⊂ Ψ (x) for every x ∈ X and σ ∈ ΣΦ(x). (4.4)

Indeed, take a nonempty finite subset σ ⊂ Φ(x). Also, for every U ∈ σ take a point q(U) ∈ ϕ(x) ∩ U and then set

Q = {q(U): U ∈ σ

}and P = p(σ) = {

p(U): U ∈ σ}.

Since p(U) ∈ U , U ∈ U, we get that P and Q are U-closed. Hence, by [19, Lemma 2.2], co(P ) and co(Q)

are O-closed because co(st(C,U)) ⊂ st(C,O), C ∈ C. Take a τ ∈ Γσ . Then, ΩU(τ ) ∩ co(P ) �= ∅ and co(P ) ⊂st(co(Q),O). Therefore, there exists an O ∈ O, with O ∩ ΩU(τ ) �= ∅ �= O ∩ co(Q) ⊂ O ∩ ϕ(x), because ϕ(x) isconvex. Since r(τ ) ∪ {O} ∈ N (O), according to (4.3), this now implies that

∅ �= O ∩ ϕ(x) ⊂⋃(

r(τ ) ∪ {O}) ⊂⋂

s(r(τ ) ∪ {O}) ⊂

⋂s(r(τ )

)

=⋂

�(τ) = ΩV(�(τ)

).

So, �(τ) ∈ Ψ (x), which completes the verification of (4.4).In what follows, for every U ∈ U, let αU : |N (U)

∣∣ → [0,1] be the corresponding affine coordinate function of U inN (U). Consider the continuous functions ξU : X → [0,1] defined by ξU = αU ◦ f , U ∈ U, where f : X → |N (U)|is the constructed selection for co [Φ]. Thus, we get a partition of unity {ξU : U ∈ U} on X. In fact, according to(4.1), we have that

∑{ξU (x): U ∈ U

} =∑{

αU(f (x)): U ∈ car(f (x))} = 1.

Hence, we can define another continuous function ξ : X → [0,1] by letting for x ∈ X that ξ(x) = max{ξU (x): U ∈ U}.Since for every x ∈ X there exists at least one U ∈ U, with ξU (x) > 0, we get that ξ : X → (0,1]. Finally, for everyx ∈ X, let

ρ(x) = {U ∈ U: ξU (x) � 2−1 · ξ(x)

}.

Note that ρ(x) ∈ Γσ for some σ ∈ ΣΦ(x). Indeed, ξ(x) > 0 and therefore, by (4.1), ρ(x) is a nonempty subset ofcar(f (x)). On the other hand, f (x) ∈ |Γσ | for some finite σ ⊂ Φ(x). Hence, ρ(x) ⊂ car(f (x)) ∈ Γσ .

We now finish the proof by taking π = � ◦ ρ : X → N (V). By (4.4), the map π is a selection for Ψ , which, by(4.2), implies (a). To see (b), take a point x ∈ X such that U \ ρ(x) �= ∅, and consider the neighborhood W of x

defined by

W = {z ∈ X: max

{ξU (z): U ∈ U \ ρ(x)

}< 2−1 · ξ(z)

}.


Then, whenever z ∈ W and U ∈ U \ ρ(x), we have that ξU (z) < 2−1 · ξ(z). Hence, U /∈ ρ(z) and therefore ρ(z) ⊂ρ(x). Clearly, this implies that

π(z) = �(ρ(z)

) ⊂ �(ρ(x)

) = π(x),

which is (b). The proof completes. �Corollary 4.2. Let X be a paracompact space, (Y,μ,C) be an S4 uniform convex structure with compact polytopesand with connected convex sets, V ∈ μ be an open convex cover of Y , and let ϕ : X � Y be an l.s.c. convex-valuedmapping. Then, ϕ has an l.s.c. convex-valued selection ψ : X � Y such that {ψ(x): x ∈ X} is a refinement of V .

Proof. By [19, Theorem 2.4], (Y,μ,C) is closure stable, while, by [19, Theorem 2.6], there are open convex coversU and O of Y such that O is a star-refinement of V , U is a refinement of O, and co(st(C,U)) ⊂ st(C,O) forevery convex set C ∈ C. Then, let π : X → N (V) be as in Lemma 4.1, and let Ω : X � Y be defined by Ω(x) =ΩV(π(x)) = ⋂

π(x), x ∈ X. Also, let ψ : X � Y be defined by ψ(x) = ϕ(x) ∩ Ω(x), x ∈ X. By (b) of Lemma 4.1,Ω has an open graph. Hence, ψ is l.s.c. because ϕ is l.s.c. It is also clear that ψ is convex-valued because both ϕ andΩ are convex-valued. This completes the proof because ψ(x) ⊂ V for every V ∈ π(x) and x ∈ X. �Proof of Theorem 1.1. Let X, Y , d and Φ be as in that theorem. For convenience, set ϕ0 = Φ . By [19, Theorem 2.6],Y has a base of uniform convex open covers. Then, inductively, by Corollary 4.2, we can construct a decreasingsequence {ϕn: n < ω} of l.s.c. convex-valued mappings ϕn : X � Y such that diamd(ϕn+1) � 1/(n + 1), n < ω. Notethat ϕn(x) ⊂ ϕ0(x) = Φ(x) for every x ∈ X and n < ω. Hence, Proposition 3.1 completes the proof. �

For a space Y , let F (Y ) be the set of all nonempty closed subsets of Y . Also, let K(Y ) denote all compact membersof F (Y ).

Recall that the Vietoris topology τV on F (Y ) is generated by all collections of the form

〈V〉 ={S ∈ F (Y ): S ⊂

⋃V and S ∩ V �= ∅, whenever V ∈ V

},

where V runs over the finite families of open subsets of Y .The following observation follows a part of the proof of [19, Theorem 4.3].

Proposition 4.3. Let (Y,C) be a uniformizable convex structure with compact polytopes, X be a space, and letψ : X � Y be an l.s.c. mapping. Define a mapping ϕ : X � Y by ϕ(x) = co(ψ(x)), x ∈ X. Then, ϕ is also l.s.c.

Proof. Take a point x0 ∈ X, and an open set O ⊂ Y , with ϕ(x0) ∩ O �= ∅. Then, there exists a finite set F ⊂ ψ(x0)

such that co(F ) ∩ O �= ∅. Since (Y,C) has compact polytopes, by [19, Theorem 2.3], the “convex hull” map co :ΣY → K(Y ) is continuous when both ΣY (the nonempty finite subsets of Y ) and K(Y ) are endowed with the relativeVietoris topology. On the other hand, 〈{O,X}〉 is a τV -neighborhood of co(F ). Hence, there exists a finite family Vof open subsets of Y such that F ∈ 〈V〉 and co(〈V〉 ∩ ΣY ) ⊂ 〈{O,X}〉. Finally, set

U =⋂{

ψ−1(V ): V ∈ V},

which is a neighborhood of x0 because ψ is l.s.c. Take a point x ∈ U , and next an yV ∈ ψ(x) ∩ V for every V ∈ V .Then, T = {yV : V ∈ V} ∈ 〈V〉 ∩ ΣY , and therefore co(T ) ∩ O �= ∅. This completes the proof because T ⊂ ψ(x), so∅ �= co(T ) ∩ O ⊂ ϕ(x) ∩ O . �

Recall that a T1-space X is τ -collectionwise normal, where τ � 2 is a cardinal number, if every discrete collectionD of closed subsets of X, with Card(D) � τ , can be separated by an open discrete collection {VD: D ∈ D} (i.e.,D ⊂ VD for every D ∈ D). Observe that a space X is normal if and only if it is 2-collectionwise normal. It is wellknown that every normal space is ω-collectionwise normal, [3].

The following consequence is a generalization of [19, Theorem 4.3(a)].

Corollary 4.4. Let X be a τ -collectionwise normal space, for some infinite cardinal τ , (Y,C) be a completelymetrizable S4 convex structure with compact polytopes and with connected convex sets, and let w(Y) � τ . Also,


let ϕ : X � Y be an l.s.c. mapping such that, for every x ∈ X, ϕ(x) is a convex compact subset of Y , or ϕ(x) = Y .Then, ϕ has a single-valued continuous selection.

Proof. By a result of [5] (see, also, [16]), there exists a metrizable space Z, a continuous map h : X → Z and an l.s.c.(compact-valued) mapping ψ : Z � Y such that ψ(h(x)) ⊂ ϕ(x), x ∈ X. Set Ψ (z) = co(ψ(z)), z ∈ Z, which is l.s.c.by virtue of Proposition 4.3 and [13, Proposition 2.3]. Also, observe that, by [19, Theorem 2.4], Ψ is convex-valued.Hence, by Theorem 1.1, Ψ admits a single-valued continuous selection g : Z → Y . Then, f = g ◦ h : X → Y is acontinuous selection for ϕ. �

To prepare for our next consequence, for a metric space (Y, d), let us recall that the Hausdorff topology τH(d) onF (Y ) is the one generated by the Hausdorff distance H(d) associate to d . Here,

H(d)(D,S) = sup{d(S, y) + d(y,T ): y ∈ S ∪ T

}, S, T ∈ F (Y ).

It should be mentioned that τV and τH(d) are, in general, not comparable, but they coincide on the subset K(Y ) ofF (Y ). In fact, τV = τH(d) if and only if Y is compact, see [11].

Corollary 4.5. Let (Y,C) be a completely metrizable S4 convex structure with compact polytopes and with connectedconvex sets, and let d be a compatible metric on Y . Then, there exists a map f : F (Y ) → Y which is continuous withrespect to both τV and τH(d), and f (S) ∈ co(S) for every S ∈ F (Y ).

Proof. Consider the d-proximal topology τδ(d) on F (Y ) [1], which is generated by all collections of the form

〈〈V〉〉 ={S ∈ 〈V〉: d

(S,Y \

⋃V

)> 0

},

where V is a finite family of open subsets of Y . It is well known that τδ(d) is coarser that both τV and τH(d), [1]. Hence,it suffices to construct a map f : F (Y ) → Y which is continuous with respect to τδ(d), and f (S) ∈ co(S) for everyS ∈ F (Y ). To this end, by [7, Theorem 1.1], there exists a metrizable space X, a map h : F (Y ) → X, continuouswith respect to τδ(d), and an l.s.c. mapping ψ : X � Y such that ψ(f (S)) ⊂ S for every S ∈ F (Y ). Then, just likein the previous proof, there exists a continuous map g : X → Y such that g(x) ∈ co(ψ(x)), x ∈ X. Now, we can takef = g ◦ h. �5. Small selections and Fréchet spaces

As it was mentioned in the Introduction, Theorem 1.1 is a natural generalization of the following Michael selectiontheorem.

Theorem 5.1. (See [14].) Let X be a paracompact space, Y be a Fréchet space, and let Φ : X � Y be an l.s.c. closedconvex valued mapping. Then, Φ admits a single-valued continuous selection.

Recall that a topological vector space is Fréchet if it is completely metrizable and locally convex.Concerning the proof of Theorem 5.1, it should be mentioned that it is based on a reduction to the classical

Michael’s selection theorem for Banach spaces [12,13]. Nevertheless this theorem was covered by Theorem 1.1,we demonstrate below a simple direct proof based on small selections.

Proof of Theorem 5.1. Take a complete metric d on Y , compatible with the topology of Y . According to Proposi-tion 3.1, it suffices to show that, for every ε > 0, every convex-valued l.s.c. mapping ϕ : X � Y has a convex-valuedl.s.c. selection ψ : X � Y , with diamd(ψ) < ε. To this end, define a c-structure : ΣY � Y on Y by (σ ) = co(σ ),σ ∈ ΣY . Also, let V ⊂ Y be an open convex set containing the origin of Y such that diamd(V ) < ε. Finally,define a mapping Ψ : X � Y by Ψ (x) = {y ∈ Y : ϕ(x) ∩ (y + V ) �= ∅}, x ∈ X, which is a Browder mapping be-cause ϕ is l.s.c. Then, by Corollary 2.3, co [Ψ ] has a continuous selection f : X → Y . Since ϕ is convex-valued,ψ(x) = ϕ(x) ∩ (f (x) + V ) �= ∅ for every x ∈ X. Hence, by [13, Proposition 2.5], ψ : X � Y is l.s.c., and clearly it isconvex-valued. Also, ψ is a selection for ϕ and diamd(ψ) � diamd(V ) < ε. The proof completes. �


Remark 5.2. The selection f in the proof of Theorem 5.1 can be obtained by [13, Lemma 4.1], which is a very simplestatement, but to finalize the proof one should take V to be a symmetric neighborhood of the origin of Y , i.e. V = −V .Namely, according to [13, Lemma 4.1], there exists a continuous map f : X → Y such that f (x) ∈ (ϕ(x)+V ), x ∈ X.Since V is symmetric, this now implies that (f (x) + V ) ∩ ϕ(x) �= ∅ for every x ∈ X.

Remark 5.3. Our proof of Theorem 5.1 is very closed to the original Michael’s arguments in [13]. In fact, the onlydifference is that the resulting selection for Φ constructed in [13] is obtained by a Cauchy sequence of continuoussingle-valued maps. Because of this, the proof in [13] involves bounded neighborhoods of the origin of Y . Recall thata subset V of a topological vector space is bounded if for every neighborhood U of the origin there corresponds a realnumber r > 0 such that V ⊂ t · U for every t > r . It is well known that a space Y is normable if and only if its originhas a convex bounded neighborhood, see [17]. Hence, the arguments in [13] were naturally restricted to completenormed spaces, i.e. Banach spaces. In this regard, our proof of Theorem 5.1 is based on small selections and convexneighborhoods of the origin of Y which are bounded with respect to a metric d on Y , hence only d-bounded, but notnecessarily bounded as subsets of the topological vector space Y .

Remark 5.4. For another direct proof of the Michael’s selection theorem in Fréchet spaces, based on Cauchy se-quences of continuous single-valued maps, we refer the interested reader to Theorem 7.1 in Chapter II of [2].

6. Small selections and metric l.c.-spaces

Suppose that Y is a space and : ΣY � Y is a c-structure on Y . In this case, the pair (Y,) was called a c-space.A nonempty subset E ⊂ Y is called a -set [9,10] if (σ ) ⊂ E for every σ ∈ ΣE . If (Y, d) is a metric space, then thetriple (Y, d,) is called a metric l.c.-space if each open ball Bd

ε (E), ε > 0, is a -set for every -set E ⊂ Y and everysingleton E = {y}, y ∈ Y .

The purpose of this section is to give an alternative proof of the following theorem.

Theorem 6.1. (See [10].) Let X be a paracompact space, (Y, d,) be a complete metric l.c.-space, and let Φ : X � Y

be an l.s.c. closed-valued mapping such that each Φ(x), x ∈ X, is a -set of Y . Then, Φ admits a single-valuedcontinuous selection.

Proof. Take an ε > 0. According to Proposition 3.1, it suffices to show that every l.s.c. “-set”-valued mappingϕ : X � Y has an l.s.c. “-set”-valued selection ψ : X � Y , with diam(ψ) � ε. To this end, define a Browder mappingΨ : X � Y by Ψ (x) = {y ∈ Y : Bd

ε (y) ∩ ϕ(x) �= ∅}, x ∈ X. By Corollary 2.3, co [Ψ ] has a continuous selectiong : X → Y . Observe that Bd

ε (g(x)) ∩ ϕ(x) �= ∅ for every x ∈ X. Indeed, since g(x) ∈ co [Ψ ](x), there exists anonempty finite σ ⊂ Ψ (x), with g(x) ∈ (σ ). By construction, σ ⊂ Bd

ε (ϕ(x)), while, by hypothesis, Bdε (ϕ(x)) is a

-set. Hence, g(x) ∈ (σ ) ⊂ Bdε (ϕ(x)), so Bd

ε (g(x)) ∩ ϕ(x) �= ∅. Now, we can define ψ : X � Y by letting ψ(x) =Bd

ε (g(x)) ∩ ϕ(x), x ∈ X. Then, by [13, Proposition 2.5], ψ is l.s.c., and each ψ(x), x ∈ X, is a -set because so areϕ(x) and Bd

ε (g(x)), x ∈ X. This completes the proof. �References

[1] G. Beer, A. Lechicki, S. Levi, S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pure Appl. 162 (1992)367–381.

[2] C. Bessaga, A. Pełczynski, Selected Topics in Infinite-Dimensional Topology, PWN, Warsaw, 1975.[3] R.H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951) 175–186.[4] F.E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968) 283–301.[5] M. Choban, S. Nedev, Factorization theorems for set-valued mappings, set-valued selections and topological dimension, Math. Balkanica 4

(1974) 457–460 (in Russian).[6] R. Engelking, General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989.[7] V. Gutev, Weak factorizations of continuous set-valued mappings, Topology Appl. 102 (2000) 33–51.[8] V. Gutev, Selections and approximations in finite-dimensional spaces, Topology Appl. 146–147 (2005) 353–383.[9] C.D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991) 341–357.

[10] C.D. Horvath, Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse 2 (2)(1993) 253–269.


[11] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951) 152–182.[12] E. Michael, Selected selections theorems, Amer. Math. Monthly 63 (1956) 233–238.[13] E. Michael, Continuous selections I, Ann. of Math. 63 (1956) 361–382.[14] E. Michael, A selection theorem, Proc. Amer. Math. Soc. 17 (6) (1966) 1404–1406.[15] J. van Mill, The Infinite Dimensional Topology of Function Spaces, North-Holland, Amsterdam, 2001.[16] S. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980) 291–317.[17] W. Rudin, Functional Analysis, second ed., McGraw–Hill Inc., New York, 1991.[18] Jr. S.B. Nadler, Hyperspaces of Sets, Monographs in Pure and Appl. Math., vol. 49, Dekker, New York, 1978.[19] M. van de Vel, A selection theorem for topological convex structures, Trans. Amer. Math. Soc. 336 (2) (1993) 463–496.[20] M. van de Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993.



Selections and totally disconnected spaces

Valentin Gutev a,∗, Tsugunori Nogura b

a School of Mathematical Sciences, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africab Department of Mathematics, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan

Received 17 May 2006; received in revised form 4 June 2007; accepted 4 June 2007

Abstract

We demonstrate that for every n < ω there exists a separable completely metrizable space Xn which has a continuous selectionfor its Vietoris hyperspace of nonempty compact subsets, but dim(Xn) = n. Related results and open problems are discussed.© 2007 Elsevier B.V. All rights reserved.

MSC: 54B20; 54C65; 54C60

Keywords: Hyperspace topology; Vietoris topology; Continuous selection; Totally disconnected space

1. Introduction

Let X be a topological space, and let F (X) be the set of all nonempty closed subsets of X. Also, let D ⊂ F (X).A map f :D → X is a selection for D if f (S) ∈ S for every S ∈ D . A selection f :D → X is continuous if it iscontinuous with respect to the relative Vietoris topology τV on D . Let us recall that τV is generated by all collectionsof the form

〈V 〉 ={S ∈ F (X): S ⊂

⋃V and S ∩ V = ∅, whenever V ∈ V

},

where V runs over the finite families of open subsets of X.In the sequel, all spaces are assumed to be at least Hausdorff. Also, for a subset D ⊂ F (X), we will use Vcs[D]

to denote the set of all Vietoris continuous selections for D . Finally, throughout this note, we will use the followingspecial subsets of F (X), where n � 1.

• Fn(X) = {S ∈ F (X): |S| � n},• [X]n = {S ⊂ X: |S| = n},• C (X) = {S ∈ F (X): S is compact}.

Suppose that f :F2(X) → X is a selection. Then, it defines a natural order-like relation � on X by letting thatx � y if and only if f ({x, y}) = x, see [9]. This relation is very similar to a linear order on X in that it is both total

* Corresponding author.E-mail addresses: [email protected] (V. Gutev), [email protected] (T. Nogura).


V. Gutev, T. Nogura / Topology and its Applications 155 (2008) 824–829 825

and antisymmetric, but, unfortunately, it may fail to be transitive. In fact, the clopen subsets of X are an indication forthe “non-transitivity” of �, see [4, Proposition 2.2]. In this regard, one of the fundamental questions in the theory ofcontinuous selections for at most 2-point subsets seems to be the following one.

Question 1. (van Mill and Wattel [11].) Let X be a space, with Vcs[F2(X)] = ∅. Does there exist a linear order � onX such that, for each y ∈ X, the sets {x ∈ X: x � y} and {x ∈ X: y � x} are both closed?

Recall that a space X is orderable (or, linearly orderable) if the topology of X coincides with the open intervaltopology on X generated by a linear ordering on X. Following [11], we say that a space X is weakly orderable ifthere exists a coarser orderable topology on X. In this terminology, Question 1 states the conjecture that a space X

is weakly orderable provided Vcs[F2(X)] = ∅. In view of that, a selection f :F2(X) → X is often called a weakselection for X. For a detailed discussion on Question 1, we refer the interested reader to [7].

In the present paper we are interested in this question from a different point of view. Namely, we are lookingfor properties that follow from orderability, and for those of them that may depend on continuous weak selections.Clearly, ind(X) � 1 for every orderable space X, where ind(X) denotes the small inductive dimension of X. On thisbase, the following question was posed in [5].

Question 2. Let X be a space, with Vcs[F2(X)] = ∅. Then, is it true that ind(X) � 1?

We are now ready to state the main purpose of this paper. In the next section, we demonstrate that the answer toQuestion 2 is in the negative by showing that, for every n < ω, there exists a separable completely metrizable space Xn

such that dim(Xn) = n and Vcs[F2(Xn)] = ∅ (Corollary 2.4). In fact, there also exists a strongly infinite-dimensionalseparable completely metrizable space X, with Vcs[F2(X)] = ∅ (Corollary 2.4). Here, dim(Y ) denotes the coveringdimension of a space Y . It should be mentioned that these results are based on natural examples of totally disconnectedspaces of arbitrary covering dimension (see, for instance, [10]), and a general selection result for totally disconnectedspaces (Theorem 2.1).

Relying on some natural relations between totally disconnected spaces and strongly zero-dimensional spaces, wefurther generalize these results by showing that, for every n < ω, there exists a separable completely metrizablespace Xn such that dim(Xn) = n and Vcs[C (Xn)] = ∅ (Proposition 3.1). Also, that there exists a strongly infinite-dimensional separable completely metrizable space X, with Vcs[C (X)] = ∅ (Proposition 3.1). However, in general,we do not know if these results can be extended to selections for F (X), hence we have the following revised versionof Question 2.

Question 3. Let X be a space, with Vcs[F (X)] = ∅. Then, is it true that ind(X) � 1?

For a topological space (X,T ), where T is the topology on X, and a family D of nonempty T -closed subsetsof X, we will sometimes write Vcs[D,T ] instead of Vcs[D] to suggest explicitly that this is the set of all selectionsfor D which are continuous with respect to the topology T on X and the Vietoris topology on D generated by T .

Going back to Question 1, if T0 is an orderable topology on X, which is coarser than the original one T , then, by[9, Lemma 7.5.1], we have that

∅ = Vcs[F2(X),T0

] ⊂ Vcs[F2(X),T

].

On the other hand, if (X,T ) is a space and T1 is a topology on X, which is finer than T , then, by [4, Corollary 3.2],Vcs[F2(X),T ] ⊂ Vcs[F2(X),T1]. Motivated by this, for a space (X,T ), a coarser Hausdorff topology T0 ⊂ T onX and a family D of nonempty T0-closed subsets of X, we shall say that T0 is selection admissible with respect to D ,or merely selection D -admissible, if ∅ = Vcs[D,T0] ⊂ Vcs[D,T ]. In these terms, a space X is weakly orderable iff ithas an orderable selection F2(X)-admissible topology. Hence, we have the following further refinement of Question 2.

Question 4. Let (X,T ) be a space, with Vcs[F2(X),T ] = ∅. Then, does there exist a selection F2(X)-admissibletopology T0 on X such that ind(X,T0) � 1?

The answer to this question is in the positive if X is connected [9, Lemma 7.2] or locally connected [12, Theorem 4and Remark 16], also when X is compact (hence, locally compact as well) [11, Theorem 1.1]. In this regard, we do not

826 V. Gutev, T. Nogura / Topology and its Applications 155 (2008) 824–829

know the answer when X is totally disconnected, even if all examples in this paper are of totally disconnected spacesX that have a zero-dimensional selection F2(X)-admissible topology. On the other hand, it is clear that a positiveanswer to Question 1 will imply a positive answer to Question 4, hence a negative answer to Question 4 will imply anegative answer to Question 1.

In conclusion, let us also mention that the paper contains other related results (see, for instance, Theorem 3.3), andsome further open questions.

2. A selection theorem for totally disconnected spaces

In this section we prove a natural selection result, and derive several applications from it. To prepare for this, for aspace X and n � 2, let

Ω(Xn

) = {(x1, x2, . . . , xn) ∈ Xn:

∣∣{x1, x2, . . . , xn}∣∣ = n

}.

Observe that if n = 2, then Ω(X2) = X2 \ Δ(X), where Δ(X) = {(x, x): x ∈ X} is the diagonal of X2.In what follows, we will use �(X) to denote the Lindelöf number of a space X. Also, for simplicity, for finitely

many open sets U0, . . . ,Uk ⊂ X, k < ω, we let

〈U0, . . . ,Uk〉 = ⟨{U0, . . . ,Uk}⟩.

Finally, recall that a space X is totally disconnected if each singleton of X is the intersection of clopen subsets of X.We are now ready to prove the following theorem.

Theorem 2.1. Let X be a totally disconnected space such that �(Ω(Xn)) � ω for some n � 2. Then, [X]n has acontinuous selection.

Proof. Since X is totally disconnected, for every α = {x1, x2, . . . , xn} ∈ [X]n there is a pairwise disjoint family{U1,U2, . . . ,Un} of clopen subsets Uk ⊂ X, 1 � k � n, such that xk ∈ Uk for every k, with 1 � k � n. Then, Uα =〈U1,U2, . . . ,Un〉 ∩ [X]n is a τV -clopen neighbourhood of α in [X]n because

Uα = [X]n ∖ ⋃{〈X \ Uk〉: 1 � k � n}.

Hence, we can define a continuous selection fα :Uα → X for Uα by letting, for instance, that

fα(β) ∈ U1 ∩ β, for every β ∈ Uα .

Thus, we get a τV -clopen cover {Uα: α ∈ [X]n} of [X]n, and a continuous selection fα for Uα , for each α ∈ [X]n.Consider the map h :Ω(Xn) → [X]n defined by h(x1, . . . , xn) = {x1, . . . , xn}, whenever (x1, . . . , xn) ∈ Ω(Xn). Then,h is a continuous surjection, hence �([X]n) � ω because �(Ω(Xn)) � ω. Therefore, there exists a countable subset{α(k) ∈ [X]n: k < ω} ⊂ [X]n such that [X]n = ⋃{Uα(k): k < ω}. For convenience, let Uk = Uα(k) and fk = fα(k),k < ω. Finally, for every α ∈ [X]n, let k(α) = min{k < ω: α ∈ Uk}. Then, define a selection f for [X]n by

f (α) = fk(α)(α), α ∈ [X]n.It only remains to show that f is continuous. Take an α ∈ [X]n, and then observe that

Fα =⋃{

Uk: k < k(α)}

is a τV -closed subset of [X]n as a finite union of τV -closed sets, while, by the definition of k(α), α /∈ Fα . Therefore,Vα = Uk(α) \ Fα is a τV -neighbourhood of α, with the property that

k(β) = k(α), for every β ∈ Vα .

That is, f � Vα = fk(α) � Vα , which completes the proof. �We now turn to some possible applications.

Corollary 2.2. Let X be a totally disconnected space, with �(Ω(X2)) � ω. Then, F2(X) has a continuous selection.In particular, Vcs[F2(X)] = ∅ for every second countable totally disconnected space X.


Proof. First of all, let us observe that F2(X) has a continuous selection if and only if [X]2 has a continuous selection,see, for instance, [5, Proposition 1.4]. On the other hand, �(Ω(X2)) � ω for every second countable space X becauseΩ(X2) is itself second countable. Hence, Theorem 2.1 completes the proof. �

It should be mentioned that Corollary 2.2 is not true if it is only supposed that �(X2) � ω. For instance, one can takeX to be the one-point compactification of an uncountable discrete space. Then, �(X2) � ω because X2 is compact, butit is well known that X has no continuous weak selection. This also demonstrates that, in Theorem 2.1, the hypothesis“�(Ω(Xn)) � ω” cannot be replaced by “�(Xn) � ω”.

For our next consequence, let us recall that the “rational Hilbert” space H , known as the Erdös space, is a separablemetrizable space which is totally disconnected, but dim(H) = 1 = dim(H × H).

Corollary 2.3. Let H be the Erdös space. Then, F2(H) has a continuous selection.

In fact, it is well known that for every n < ω there exists a separable completely metrizable space Xn which istotally disconnected, but dim(Xn) = n. Also, that there exists a strongly infinite-dimensional separable completelymetrizable space X which is totally disconnected [8] (see also [10, Theorem 4.7.10]). Hence, we have also the follow-ing immediate consequence.

Corollary 2.4. For every n < ω there exists a separable completely metrizable space Xn such that F2(Xn) has a con-tinuous selection, but dim(Xn) = n. Also, there exists a strongly infinite-dimensional separable completely metrizablespace X such that F2(X) has a continuous selection.

An interesting direction to improve Theorem 2.1 is to look for a possible extension of the domain of the selectionconstructed in this theorem. In this regard, we have the following particular question.

Question 5. Let X be a second countable totally disconnected space. Does there exist a continuous selection forC (X)? In particular, does there exist a continuous selection for Fn(X) for every n � 3?

Related to this question, we have the following partial result.

Corollary 2.5. Let X be a second countable totally disconnected space. Then, there exists a continuous selection forF4(X).

Proof. According to Theorem 2.1, Corollary 2.2, and [3, Corollary 4.1], there exists a continuous selection for F3(X)

because Vcs[F2(X)] = ∅ = Vcs[[X]3]. Hence, by [6, Theorem 4.1], F4(X) has a continuous selection as well. �3. Selections and zero-dimensional spaces

The selection construction in the proof of Theorem 2.1 suggests some possible relations with (strongly) zero-dimensional spaces. To this end, to any totally disconnected topology T on X we will associate the topology Tclopengenerated by the clopen members of T . Clearly, (X,Tclopen) is a zero-dimensional space and Tclopen ⊂ T . Hence, theidentity map idX : (X,T ) → (X,Tclopen) is a continuous bijection. Thus, �(X,Tclopen) � ω provided �(X,T ) � ω,and, therefore, (X,Tclopen) is Lindelöf provided so is (X,T ). In particular, this implies that (X,Tclopen) is stronglyzero-dimensional if (X,T ) is Lindelöf [14] (see also [1, Theorem 6.2.7]).

In fact, the examples of second countable totally disconnected spaces (X,T ) constructed in [10, Theorem 4.7.10]have also the property that (X,Tclopen) is separable and metrizable (hence, second countable as well). In view of that,we have the following further result, which improves Corollary 2.4.

Proposition 3.1. For every n < ω there exists a separable completely metrizable space Xn such that C (Xn) has a con-tinuous selection, but dim(Xn) = n. Also, there exists a strongly infinite-dimensional separable completely metrizablespace X such that C (X) has a continuous selection.

828 V. Gutev, T. Nogura / Topology and its Applications 155 (2008) 824–829

Proof. Suppose that (Z,T ) is a totally disconnected space such that (Z,Tclopen) is a separable metrizable space.Then, being zero-dimensional, (Z,Tclopen) is a subset of the Cantor set [13] (see also [2, Theorem 1.3.15]). Hence,there exists a linear order � on Z such that (Z,T ) is weakly orderable with respect to this order, and every nonemptycompact subset of (Z,T ) has a “�”-first element. Therefore, by [9, Lemma 7.5.1], C (Z) has a continuous selection.To finish the proof, it now remains to observe that, by [10, Theorem 4.7.10], for every n < ω there exists a totallydisconnected separable completely metrizable space (Xn,T ) such that dim(Xn,T ) = n and (Xn,Tclopen) is separa-ble and metrizable; by the same theorem, there exists a totally disconnected strongly infinite-dimensional separablecompletely metrizable space (X,T ) such that (X,Tclopen) is separable and metrizable as well. �

The proof of Proposition 3.1 suggests a possible approach to construct selections for totally disconnected spacesbased on zero-dimensionality provided the answer to the following question is in the positive.

Question 6. Let (X,T ) be a second countable totally disconnected space. Then, is (X,Tclopen) a second countablespace?

Related to selections, we have the following consequence of Theorem 2.1 which may shed some light on thisquestion.

Corollary 3.2. Let (X,T ) be a totally disconnected space such that, with respect to the product topology on Ω(X2)

generated by T , the Lindelöf number of Ω(X2) is countable. Then, Vcs[F2(X),Tclopen] = ∅. In particular, Tclopenis a selection F2(X)-admissible topology on X.

Proof. As mentioned before, the identity map idX : (X,T ) → (X,Tclopen) is a continuous bijection. Hence, the iden-tity map idΩ(X2) :Ω(X2) → Ω(X2) is also a continuous bijection when the domain is endowed with the producttopology generated by T and the range is endowed with the one generated by Tclopen. This implies that the Lin-delöf number of Ω(X2) is also countable when it is endowed with the product topology generated by Tclopen. Then,Corollary 2.2 and [4, Corollary 3.2] complete the proof. �

We conclude this paper by showing that Vcs[C (X),Tclopen] ⊂ Vcs[C (X),T ] for any totally disconnected space(X,T ), where C (X) denotes the compact subsets of X with respect to T . In fact, we demonstrate a little bit more byproving the following natural generalization of [4, Corollary 3.2]. Recall that we consider only Hausdorff topologicalspaces, hence all topologies are assumed to be at least Hausdorff.

Theorem 3.3. Let (X,T ) be a space, T0 ⊂ T be a coarser topology on X, and let D be a nonempty family consistingof nonempty T -compact subsets of X. Then, Vcs[D,T0] ⊂ Vcs[D,T ]. In particular, T0 is a selection D -admissibletopology on X if Vcs[D,T0] = ∅.

Proof. Take a selection f ∈ Vcs[D,T0], an S ∈ D , and a T -neighbourhood U of f (S) in X. If S ⊂ U , thenf (〈U 〉 ∩ D) ⊂ U , so f will be continuous at S with respect to T for that choice of the neighbourhood U . Sup-pose that D = S \U = ∅. Then D is a T0-compact subset of X (being T -compact) and f (S) /∈ D. Hence, there existsan O ∈ T0 such that f (S) ∈ O and R ∩ D = ∅, where R is the closure of O in (X,T0). This is possible because(X,T0) is Hausdorff and D is T0-compact.

Now, on the one hand, there exists a finite family V of T0-open subsets of X such that

S ∈ 〈V 〉 and f(〈V 〉 ∩ D

) ⊂ O, (1)

because f ∈ Vcs[D,T0]. On the other hand, B = {U,X \ R} is family of T -open sets such that

S ∈ 〈B〉. (2)

Then, consider the family

W = {V ∩ B: V ∈ V , B ∈ B and V ∩ B ∩ S = ∅}. (3)

Clearly, W refines both V and B, while each element of V contains some element of W because, by (1) and (2),S ∈ 〈V 〉 ∩ 〈B〉. By the same reason, S ∈ 〈W 〉. Thus, W is a finite family of T -open sets, with S ∈ 〈W 〉 ⊂ 〈V 〉. We


are going to show that f (〈W 〉 ∩ D) ⊂ U . Indeed, take a T ∈ 〈W 〉 ∩ D . Then, T ∈ 〈W 〉 ⊂ 〈V 〉, and therefore, by (1),f (T ) ∈ O ⊂ R. However f (T ) ∈ W for some W ∈ W because f (T ) ∈ T ⊂ ⋃

W , while, by (3), W ⊂ U for everyW ∈ W , with W ∩ R = ∅. Thus, f (T ) ∈ U which completes the proof. �

By taking D = C (X) to be the nonempty T -compact subsets of X, we get the following immediate consequence.

Corollary 3.4. Whenever (X,T ) is a totally disconnected space, we have that Vcs[C (X),Tclopen] ⊂ Vcs[C (X),T ].In particular, Tclopen is a selection C (X)-admissible topology if Vcs[C (X),Tclopen] = ∅.

By taking D = Fn(X), n � 2, we get also the following consequence.

Corollary 3.5. Whenever (X,T ) is a totally disconnected space and n � 2, we have that Vcs[Fn(X),Tclopen] ⊂Vcs[Fn(X),T ]. In particular, Tclopen is a selection Fn(X)-admissible topology if Vcs[Fn(X),Tclopen] = ∅.

Corollaries 3.4 and 3.5 suggest the following two interesting questions.

Question 7. Does there exist a (second countable) totally disconnected space (X,T ) such that Vcs[C (X),T ] \Vcs[C (X),Tclopen] = ∅?

Question 8. Does there exist a (second countable) totally disconnected space (X,T ) such that Vcs[Fn(X),T ] \Vcs[Fn(X),Tclopen] = ∅ for some n � 2?

Acknowledgements

The authors would like to express their best gratitude to the referee for several valuable remarks concerning theproof of Theorem 3.3. Research of the first author is supported in part by the NRF of South Africa.

References

[1] R. Engelking, General Topology, Revised and Completed Edition, Heldermann-Verlag, Berlin, 1989.[2] R. Engelking, Theory of Dimensions: Finite and Infinite, Heldermann-Verlag, Lemgo, 1995.[3] S. García-Ferreira, V. Gutev, T. Nogura, Extensions of 2-point selections, New Zealand J. Math. (2006), in press.[4] V. Gutev, T. Nogura, Selections and order-like relations, Appl. Gen. Topol. 2 (2001) 205–218.[5] V. Gutev, T. Nogura, Some problems on selections for hyperspace topologies, Appl. Gen. Topol. 5 (1) (2004) 71–78.[6] V. Gutev, T. Nogura, Weak selections and flows in networks, preprint, 2005.[7] V. Gutev, T. Nogura, Selection problems for hyperspaces, in: E. Pearl (Ed.), Open Problems in Topology, vol. 2, Elsevier, Amsterdam, 2007,

pp. 161–170.[8] S. Mazurkiewicz, Sur les problèmes κ et λ de Urysohn, Fund. Math. 10 (1927) 311–319.[9] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951) 152–182.

[10] J. van Mill, Infinite Dimensional Topology, Prerequisites and Introduction, North-Holland, Amsterdam, 1989.[11] J. van Mill, E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (3) (1981) 601–605.[12] T. Nogura, D. Shakhmatov, Characterizations of intervals via continuous selections, Rend. Circ. Mat. Palermo, Serie II 46 (1997) 317–328.[13] W. Sierpinski, Sur les ensembles connexes et non connexes, Fund. Math. 2 (1921) 81–95.[14] N. Vedenisoff, Remarks on the dimension of topological spaces, Moskov. Gos. Univ. Uc. Zap. 30 (1939) 131–140 (in Russian).



Topological convexities, selections and fixed points

Charles D. Horvath

Département de Mathématiques, Université de Perpignan, 66860 Perpignan cedex, France

Received 28 September 2006; received in revised form 4 January 2007; accepted 4 January 2007

Abstract

A convexity on a set X is a family of subsets of X which contains the whole space and the empty set as well as the singletonsand which is closed under arbitrary intersections and updirected unions. A uniform convex space is a uniform topological spaceendowed with a convexity for which the convex hull operator is uniformly continuous. Uniform convex spaces with homotopicallytrivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizablethen a continuous selection theorem à la Michael holds. Upper semicontinuous maps have approximate selections and fixed points,under the usual assumptions.© 2007 Elsevier B.V. All rights reserved.

Keywords: Uniform spaces; Generalized convexity; Continuous selections; Fixed points

1. Introduction

A convexity on a set X is an algebraic closure operator A �→ �A� from P(X) to P(X) such that �{x}� = {x} forall x ∈ X or, equivalently, a family of subsets C of X, the convex sets, which contains the whole space and the emptyset as well as the singletons and which is closed under arbitrary intersections and updirected unions. Uniform convexspaces, that is uniform topological spaces for which the convex hull operator is a uniformly continuous map, areintroduced in Section 2.1. The definition is slightly different from that given by Van de Vel in [26]; the language ofentourages is more natural and more suggestive when one is concerned with multivalued maps than the language ofcoverings.

In [26] Van de Vel showed that a complete metric space endowed with a convexity which is compatible withthe metric, which has the Kakutani Property (i.e. disjoint convex sets C1 and C2 can be enlarged to disjoint convexsets D1 and D2 such that D2 = X \D1), and for which polytopes (i.e. sets of the form �A� where A is a finite set), arecompact and connected, is an absolute retract (Theorem 5.1 in [26]). That result is derived from a selection theoremà la Michael, assuming of course that the metric is complete. We show in Section 4 that a uniform convex space forwhich polytopes are homotopically trivial is an absolute extensor for the class of metric spaces; if it is metrizable it isof course an absolute retract. Michael’s Selection Theorem is proved in Section 3.2 for completely metrizable uniformspaces with homotopically trivial polytopes. From part (2) of Corollary 3.2 one can see that Van de Vel’s Theoremfollows from those results.



C.D. Horvath / Topology and its Applications 155 (2008) 830–850 831

In Van de Vel’s paper compactness and connectedness of polytopes, coupled with the Kakutani Property, implyan important property of nerves of coverings of convex sets by open convex sets; it is a minor modification of thatresult that we call the Van de Vel Property. The Van de Vel Property is defined in Section 2.2. Section 3.3 deals withapproximate selections for upper semicontinuous maps. Section 5 deals with the fixed point property for single valuedand multivalued maps and related intersection properties.

Although selection theorems for lower semicontinuous maps, selection theorems for upper semicontinuous maps,fixed point theorems and other related results do hold under different sets of hypotheses there is a framework (cor-responding to that of locally convex topological vector spaces) under which all of those results hold (with completemetrizability added for continuous selections of lower semicontinuous maps). That framework is in a sense a minimalmaximal framework; maximal from the point of view of the results that hold in such a context, all of the results alludedto above, and minimal from the point of view of the hypotheses which are at the same time simple and trivially truein the classical setting of topological vector spaces. The exact meaning of that discussion, which is the subject matterof the following pages, is encapsulated in the following statement.

Metatheorem. Let (X,U,C) be a uniform convex space such that:

(1) polytopes are compact metrizable and connected;(2) the Kakutani Property holds on polytopes equipped with the induced convexity.

Then,

(A) (Dugundji’s Theorem) X is an absolute extensor for the class of metric spaces; and, consequently, an absoluteretract if it is metrizable.

(B) (Michael’s Selection Theorem) if the uniformity U is completely metrizable then lower semicontinuous mapsΩ : Y → X defined on a paracompact space Y with nonempty closed convex values (i.e. Ωy ∈ C for all y ∈ Y)

have continuous selections.(C) (Approximate selections for usc maps) If Ω : Y → X is an upper semicontinuous map with nonempty convex

values defined on a paracompact space Y then, for all elements R of the uniformity U , there exists a continuousmap f : Y → X such that, for all y ∈ Y , f (y) ∈ R(Ωy). Furthermore, if the values of Ω are convex and compactthen any neighborhood Θ ⊂ Y × X of the graph of Ω contains the graph of a continuous map f : Y → X.

(D) (Fan–Himmelberg’s Theorem) If the Kakutani Property holds then compact upper semicontinuous maps Ω : X →X (i.e. the closure of Ω(X) is compact) with nonempty compact convex values have fixed points.

The paper ends with a few examples of uniform convex spaces. All spaces are assumed to be separated.

2. Convexities

2.1. Uniform convex spaces

Definition 2.1. A convexity on a set X is a family C of subsets of X such that:

(Conv1) ∅ and X belong to C and, for all x ∈ X, {x} ∈ C;(Conv2) if A is a subfamily of C then

⋂A belongs to C;

(Conv3) if A is an updirected subfamily of C then⋃

A belongs to C.

Elements of C are called convex sets and (X,C) is a convex space.

Given A ⊂ X let C(A) = {C ∈ C: A ⊂ C}; by Conv1 C(A) �= ∅ and by Conv2,⋂

C(A) ∈ C; it is the C-hull of A

and it is denoted by �A�C . Whenever a single convexity is under discussion we will drop the prefix, or the index, C.

Given a nonempty set S we will denote by 〈S〉 the family of nonempty finite subsets of S. An algebraic closureoperator on X is a map A �→ �A� from the power set of X to itself such that, for all A,B ⊂ X:

832 C.D. Horvath / Topology and its Applications 155 (2008) 830–850

(Alg1) A ⊂ �A�;(Alg2) ��A��= �A�;(Alg3) if A ⊂ B then �A� ⊂ �B�;(Alg4) �A� = ⋃{�S�: S ∈ 〈A〉}.

If C is a convexity on X then A �→ �A�C is an algebraic closure operator on X and C = {C ⊂ X: �C�C = C};reciprocally, if A �→ �A� is an algebraic closure operator on X such that ∅ = �∅� and {x} = �{x}� for all x ∈ X then{C ⊂ X: �C�= C} is a convexity on X, call it C, for which �·�= �·�C .

If C is a convexity on X then, for all convex subset C of X, C|C = {C′ ∈ C: C′ ⊂ C} is a convexity on C, theinduced convexity, and, for all A ⊂ C, �A�C|C = �A�C .

Definition 2.2. A uniform convex space is a triple (X,U,C) where U is a separated uniformity on X and C is aconvexity such that:

(UnifConv) for all R ∈ U there exists S ∈ U such that, for all C ∈ C, �S(C)� ⊂ R(C).

If (X,U,C) is a uniform convex space we say that the uniformity U and the convexity C are compatible.

Lemma 2.1. If (X,U,C) is a uniform convex space then, for all convex sets C ∈ C the induced uniformity and theinduced convexity are compatible.

Proof. The induced uniformity on a convex set C ∈ C is U |C = {R ∩C ×C: R ∈ U}; let us see that (C,U |C,C|C) is auniform convex space if (X,U,C) is one: let R and S be two elements of U such that �S(C′)� ⊂ R(C′) for all C′ ∈ C;from S(C′) ∩ C ⊂ �S(C′)�∩ C and the convexity of �S(C′)�∩ C we have �S(C′) ∩ C�⊂ R(C′) ∩ C. �

As one can easily see, locally convex topological vector spaces are uniform convex spaces, and therefore, so areconvex subsets of locally convex topological vector spaces. Uniform convex spaces are locally convex.

Proposition 2.1. In a uniform convex space (X,U,C) the closure of a convex set is convex. Furthermore, for all x ∈ X

and for all neighborhood W of x there exists a convex neighborhood V of x such that V ⊂ W.

Proof. For each R ∈ U choose R′ ∈ U such that, for all C ∈ C, �R′(C)� ⊂ R(C); from C ⊂ R′(C) ⊂ �R′(C)� andfrom C = ⋂{R(C): R ∈ U} ⊂ ⋂{R′(C): R ∈ U} we get C = ⋂{�R′(C)�: R ∈ U}.

If W is a neighborhood of x there exists R ∈ U such that Rx ⊂ W ; let V = �R′x�, it is a neighborhood of x sinceit contains R′x and, by construction, V ⊂ Rx ⊂ W . �

Let Oconv be the set of elements of U such that, for all x ∈ X, Rx is open and belongs to C.

The second part of Proposition 2.1 is strengthened by the first part of the next proposition, which is essentiallyTheorem 2.6 of Van de Vel [26].

Proposition 2.2. In a uniform convex space (X,U,C) the following properties hold:

(a) for all R ∈ U there exists S ∈ U such that, for all C ∈ C, there exists an open convex set V such that S(C) ⊂ V ⊂R(C);

(b) Oconv is a base of the uniformity U .

Proof. Let R0 = R; if Rn has been defined, choose Rn+1 ∈ U such that Rn+1 ◦ Rn+1 ⊂ Rn and choose Sn ∈ U suchthat, for all C ∈ C, �Sn(C)� ⊂ Rn+1(C). Put C0 = C and Cn+1 = �Sn(Cn)�.

We have Cn+1 ⊂ Rn+1(Cn) and therefore Rn+1(Cn+1) ⊂ Rn(Cn). From Cn ⊂ Sn(Cn) ⊂ Cn+1 we have Cn ⊂int Cn+1 ⊂ Cn+1. It follows from this that V = ⋃

n∈NCn is open and convex.

We also have S0(C) ⊂ �S0(C)� = C1 ⊂ V ; let S = S0.


Finally, we have seen that {Rn(Cn)}n∈N is a decreasing sequence of sets, and Cn ⊂ Rn(Cn) by construction; fromthis we have Cn ⊂ R0(C0) for all n ∈ N, and consequently, V ⊂ R(C). This proves (a).

To prove (b), choose, for all R ∈ U an SR ∈ U , and for all x ∈ X an open convex set VRx such that SRx ⊂ VRx ⊂Rx; from SR ⊂ VR we have VR ∈ Oconv and, from SR ⊂ VR ⊂ R, we see that {VR: R ∈ U} is a base of U . �Corollary 2.1. If (X,U,C) is a uniform convex space then, for all compact convex set K ⊂ X and for all open setU ⊂ X containing K there exists an open convex set V such that K ⊂ V ⊂ U .

Proof. For all x ∈ K there exists Sx ∈ U such that x ∈ Sxx ⊂ U ; for all x ∈ K choose Rx ∈ U such that Rx ◦Rx ⊂ Sx .By compactness there exists a finite subset {x1, . . . , xm} ⊂ K such that K ⊂ ⋃m

i=1 Rxixi ; choose R ∈ U such that R ⊂⋂m

i=1 Rxi. By (a) of Proposition 2.2 there exists an open convex set V such that K ⊂ V ⊂ R(K). For all x′ ∈ R(K)

there exists x ∈ K and i ∈ {1, . . . ,m} such that x′ ∈ Rx and x ∈ Rxixi , from which we obtain x′ ∈ (R ◦ Rxi

)xi ⊂(Rxi

◦ Rxi)xi ⊂ Sxi

xi ⊂ U . We have shown that V ⊂ U . �The first part of Proposition 2.2 will be important for the approximation of upper semicontinuous maps while the

second part will be used in the proof of the main theorem of Section 3.As a straightforward consequence of Proposition 2.2 we have:

Corollary 2.2. In a uniform convex space, a compact convex set is the intersection of the open convex sets containingit.

2.2. The Van de Vel Property

Given a set X, a reflexive relation R ⊂ X × X and a nonempty subset F of X let, for all nonempty subsets C of X,

SR(C;F) ={A ∈ 〈F 〉: C ∩

( ⋂

x∈A

Rx

)�= ∅

}.

SR(C;F) is a simplicial complex whose geometric realization is denoted by |SR(C;F)|; SR(C) stands for SR(C;X).The following sections will show that a uniform convexity for which there are enough entourages R for which thepolyhedra SR(C), where C is convex, are homotopically trivial behaves, with respect to selections and fixed pointproperties, like the linear convexity of a locally convex, or a normed, topological vector space. Under natural condi-tions, in the sense that they trivially hold for locally convex vector spaces, we will see, following essentially Van deVel, that the polyhedra SR(C) are homotopically trivial. More surprising maybe is that from the fact that polyhedraof the form SR(C) are homotopically trivial, when R is taken in a suitable basis of the uniformity, we obtain that theconvex sets themselves are homotopically trivial. We start with the following simple observation.

Lemma 2.2. Let K be a compact set in a uniform topological space (X,U) and B a basis of the uniformity U . Then|SR(K)| is connected for all R ∈ B if and only if K is connected.

Proof. Assume that K is compact and not connected. There are two open sets U1 and U2 such that K ⊂ U1 ∪ U2,K ∩ U1 ∩ U2 = ∅, K ∩ Ui �= ∅ for i = 1,2; for x ∈ K ∩ Ui we find Si

x ∈ B such that Sixx ⊂ K ∩ Ui . Proceeding as

in the proof of Corollary 2.1 we find a finite set A = {x1, . . . , xn} ⊂ K and R ∈ B such that for all x ∈ K there existsy ∈ A such that Rx ⊂ S1

yy or Rx ⊂ S2yy which implies that for all x ∈ K we have either Rx ⊂ U1 or Rx ⊂ U2. Since

K ∩ U1 ∩ U2 = ∅, |SR(K)| is not connected.Assume that K is connected and let {x} and {y} be two vertices of |SR(K)|. Choose x′ ∈ K ∩Rx and y′ ∈ K ∩Ry;

there is a finite sequence {x0, . . . , xm} of points of X such that x′ ∈ Rx0, y′ ∈ Rxm and K ∩ Rxi ∩ Rxi+1 �= ∅ fori = 0, . . . ,m − 1. We also have K ∩ Rx ∩ Rx0 �= ∅ and K ∩ Ry ∩ Rxm �= ∅. By construction, the sets {x, x0},{xi−1, xi} and {xm,y} belong to SR(K), this shows that the 1-skeleton of |SR(K)| is connected. �Definition 2.3. A uniform convex space (X,U,C) has the Van de Vel Property if there exists a base Uconv of theuniformity U such that:


(1) Uconv ⊂ Oconv and(2) for all nonempty C ∈ C and all R ∈ Uconv, the polyhedron |SR(C)| is homotopically trivial.(3) It is a Van de Vel space if for all convex sets C ∈ C, for all R ∈ Uconv and all E ⊂ X such that {Rx: x ∈ E} covers

C the polyhedron |SR(C;E)| is homotopically trivial.

Clearly, a uniform convex space which is a Van de Vel space has the Van de Vel Property. The following propositionshows that in a Van de Vel space the Van de Vel Property is hereditary with respect to convex sets equipped with theinduced uniformity and the induced convexity.

Proposition 2.3. If (X,U,C) is a Van de Vel space then, for all C ∈ C, the uniform convex space (C,U |C,C|C) hasthe Van de Vel Property.

Proof. Let Uconv be a basis of U for which (1) and (3) of Definition 2.3 are verified and let C ⊂ X be a convex set. Asone can easily see the family Uconv|C = {R ∩ (C × C): R ∈ Uconv} is a basis of the induced convexity whose elementshave convex and open (in C), values. Let Q ⊂ C be convex and pick S ∈ Uconv|C . Then S = R ∩ (C × C) for someR ∈ Uconv. For all finite subsets A of C one has Q ∩ (

⋂x∈A Sx) = Q ∩ (

⋂x∈A Rx ∩ C) = (Q ∩ C) ∩ (

⋂x∈A Rx) =

Q∩ (⋂

x∈A Rx). In other words, SS(Q) = SR(Q;C), where the first simplicial complex is computed in (C,U |C,C|C)

and the second in (X,U,C). By hypothesis, SR(Q;C) is homotopically trivial; this completes the proof. �Lemma 2.3. Let R ∈Oconv where (X,U,C) is a uniform convex space.

(1) If polytopes are compact then for all C ∈ C, for all nonempty subsets E of X such that {Rx: x ∈ E} covers C

and for all compact subsets K of |SR(C;E)| there exists a polytope P ⊂ C and a finite set F ⊂ E such thatK ⊂ |SR(P ;F)| ⊂ |SR(C;E)|.

(2) If C ∈ C and if C ⊂ ⋃x∈F Rx with F finite then there exists a polytope P ⊂ C such that |SR(C;F)| =

|SR(P ;F)|.

Proof. (1) A compact subset K of |SR(C;E)| is contained in a finite polyhedron; there is a finite subset F of E

such that K ⊂ |{A ∈ 〈F 〉 ∩ SR(C;E)}|. For each A ∈ 〈F 〉 ∩ SR(C;E) choose a point pA in C ∩ (⋂

x∈A Rx) and letP = �pA: A ∈ 〈F 〉 ∩ SR(C;E)�; P is a polytope which is contained in C; P being compact by hypothesis, there is afinite subset F0 of E such that P ⊂ ⋃

x∈F0Rx. Let F1 = F ∪ F0; since {A ∈ 〈F 〉 ∩ SR(C;E)} ⊂ SR(P,F1) we have

K ⊂ |SR(P,F1)| ⊂ |SR(C;E)|.(2) For all J ∈ SR(C;F) choose a point xJ ∈ C ∩ (

⋂x∈J Rx) and let P = �{xJ : J ∈ SR(C;F)}�; it is a polytope

contained in C, therefore SR(P ;F) ⊂ SR(C;F). If J ∈ SR(C;F) then xJ ∈ P ∩ (⋂

x∈J Rx), and therefore J ∈SR(P ;F). We have shown that SR(P ;F) = SR(C;F), and therefore |SR(P ;F)| = |SR(C;F)|. �Lemma 2.4. Let R ∈ Oconv where (X,U,C) is a uniform convex space. Statements (1) and (2) below are equivalent.If polytopes are compact then (3) is equivalent to (1) and (2).

(1) For all C ∈ C and all F ∈ 〈X〉 such that {Rx: x ∈ F } covers C |SR(C;F)| is homotopically trivial.(2) For all polytope P ⊂ X and all F ∈ 〈X〉 such that {Rx: x ∈ F } covers P , |SR(P ;F)| is homotopically trivial.(3) For all C ∈ C and all nonempty subset E of X such that {Rx: x ∈ E} covers C, |SR(C;E)| is homotopically

trivial.

Proof. The equivalence of (1) and (2) is a consequence of the second part of Lemma 2.3. To prove (3) from (1) and(2) assuming that polytopes are compact recall that a polyhedron is homotopically trivial if each of its compact subsetis contained in a homotopically trivial subset. The conclusion follows from (1) of Lemma 2.3. �

Definition 2.3 is rather abstract and one might wonder how one can check that a given uniform convex space is aVan de Vel space. Surprisingly, under very simple natural conditions—compactness and connectedness of polytopes,and the Kakutani Property—we have a Van de Vel space. This was essentially established by Van de Vel in [26] usinga different language; actually, we will prove this with a condition on polytopes formally weaker than the Kakutani


separation property. In the classical linear setting, the last condition is implied by the algebraic Hahn–Banach Theo-rem; also, in linear spaces, polytopes are compact and connected. As a consequence, an arbitrary convex subset of alocally convex topological vector space is a Van de Vel space.

Proposition 2.4. A uniform convex space (X,U,C) for which conditions (1) and (2) below hold is a Van de Vel space.

(1) Polytopes are compact and connected.(2) (S4-weak) For all polytopes P and for all disjoint nonempty open convex subsets W1 and W2 of P there exist

convex subsets C1 and C2 of P such that C1 ∩ W2 = C2 ∩ W1 = ∅ and P = C1 ∪ C2 (clearly, one can assumethat C1 and C2 are closed in P ).

Proof. Let P ⊂ X be a polytope, R an element of Oconv and F ⊂ X a nonempty finite subset such that {Rx: x ∈ F }covers C. We show by induction on the number of simplexes of the finite polyhedron |SR(P ;F)| that it is homo-topically trivial; if that number is one there is nothing to prove. Assume that number is m + 1 with m � 1; if thereis only one maximal simplex, again there is nothing to prove, otherwise choose two maximal elements A1 and A2of SR(P ;F). We have P ∩ (

⋂x∈Ai

Rx) �= ∅, i = 1,2, and P ∩ (⋂

x∈A1∪A2Rx) = ∅, since A1 ∪ A2 /∈ SR(P ;F); let

Wi = P ∩ (⋂

x∈AiRx) �= ∅. By (S4-weak) we can find two convex subsets C1 and C2 such that C1 ∩W2 = C2 ∩W1 = ∅

and P = C1 ∪ C2, and therefore SR(P ;F) = SR(C1;F) ∪ SR(C2;F).Notice that Ai ∈ SR(Ci;F), since Ci ⊂ ⋃

x∈F Rx and Ci ∩ (⋂

x∈AiRx) = Ci ∩ P ∩ (

⋂x∈Ai

Rx) = Ci ∩ Wi .Also, from C2 ∩ (

⋂x∈A1

Rx) = C2 ∩ P ∩ (⋂

x∈A1Rx) = C2 ∩ W1 = ∅ we see that A1 /∈ SR(C2;F); and, similarly,

A2 /∈ SR(C1;F). This shows that |SR(C1;F)| and |SR(C2;F)| have, each, at most m simplexes and are, by theinduction hypotheses, homotopically trivial.

Next, we show that SR(C1 ∩ C2;F) = SR(C1;F) ∩ SR(C2;F); one inclusion being trivial, we only have to seethat SR(C1;F) ∩ SR(C2;F) ⊂ SR(C1 ∩ C2;F). As we have seen, we can assume that C1 and C2 are closed in P ,and consequently, P being connected, we must have C1 ∩ C2 �= ∅.

Let A ∈ 〈F 〉 such that Ci ∩ (⋂

x∈A Rx) �= ∅, i = 1,2. The set Q = P ∩ (⋂

x∈A Rx) is connected since it is convexand Ci ∩ Q is nonempty and closed in Q; from P = C1 ∪ C2 we have Q = (C1 ∩ Q) ∪ (C2 ∩ Q), and, from theconnectedness of Q, Q ∩ C1 ∩ C2 �= ∅; this shows that A ∈ SR(C1 ∩ C2;F).

Also, neither A1 nor A2 are in SR(C1 ∩ C2;F); by the induction hypotheses, |SR(C1 ∩ C2;F)| is homotopicallytrivial.

In conclusion, the finite polyhedron |SR(P ;F)| is the union of two homotopically trivial polyhedra |SR(C1;F)|and |SR(C2;F)| whose intersection is also homotopically trivial. Since finite homotopically trivial polyhedra areabsolute retracts, and |SR(C1;F)| ∩ |SR(C2;F)| = |SR(C1 ∩ C2;F)| we have shown that |SR(P ;F)| is the unionof two absolute retracts whose intersection is an absolute retract, it is therefore an absolute retract and consequentlyhomotopically trivial. �Corollary 2.3. Convex subsets of locally convex vector spaces are Van de Vel spaces.

In all the propositions and proofs that follow, Uconv will always stand for a basis of the uniformity U of a givenuniform convex space (X,U,C) for which (1) and (2) of Definition 2.3 holds.

3. Selections for lower semicontinuous maps

3.1. Selections for lower semicontinuous maps defined on finite dimensional spaces

Given a map Ω : Y → X and a binary relation R ⊂ X we say that Ω is R-small if for all y ∈ Y there exists x ∈ X

such that Ωy ⊂ Rx.

Theorem 3.1. A lower semicontinuous map with nonempty (closed) convex values Ω : Y → X from a finite dimen-sional paracompact topological space to a uniform convex space (X,U,C) with the Van de Vel Property has, for allR ∈ Uconv, a lower semicontinuous selection ΓR : Y → X with (closed) convex values which is R-small.


Theorem 3.2. If (X,U,C) is a completely metrizable uniform convex space with the Van de Vel Property then anylower semicontinuous map Ω : Y → X with closed and nonempty convex values defined on a paracompact finitedimensional space Y has a continuous selection.

The proof of Theorem 3.1 will proceed as follows: given R ∈ Uconv we will show that there exists a map ω : Y →〈X〉 such that

(1) for all y ∈ Y Ωy ∩ (⋂

x∈ω(y) Rx) �= ∅;(2) for all y ∈ Y there exists a neighborhood V of y such that, for all y′ ∈ V , ω(y′) ⊂ ω(y).

We will then put ΩRy = Ωy ∩ (⋂

x∈ω(y) Rx); by (1), ΩRy is convex and nonempty. If U ⊂ X is open and ΩRy ∩U �= ∅ then, taking into account that Ω is lower semicontinuous and that

⋂x∈ω(y) Rx is open we find a neighborhood

W of y such that, for all y′ ∈ W , Ωy′ ∩ (⋂

x∈ω(y) Rx) ∩ U �= ∅; next, taking V as in (2) above, we will have Ωy′ ∩(⋂

x∈ω(y′) Rx) ∩ U �= ∅ for all y′ ∈ V ∩ W , which shows that ΩR is lower semicontinuous; finally, we take ΓRy =ΩRy.

That there is a map ω : Y → 〈X〉 for which (1) and (2) hold is a consequence of Lemma 3.1 below.Given Ω : Y → X, y ∈ Y and R ∈ Uconv, let, as previously, SR(Ωy) = {A ∈ 〈X〉: Ωy ∩ (

⋂x∈A Rx) �= ∅} and put

SR = {A ∈ 〈X〉: ⋂x∈A Rx �= ∅}. |SR(Ωy)| is a subpolyhedron of |SR| which is itself a subpolyhedron of |〈X〉|; we

will denote by |SR(Ω)| the map y �→ |SR(Ωy)|.

Lemma 3.1. If Ω : Y → X is a lower semicontinuous map with nonempty convex values from a finite dimensionalparacompact topological space to a uniform convex space (X,U,C) with the Van de Vel Property then, for all R ∈Uconv, the map |SR(Ω)| : Y → |SR| has a continuous selection.

The proof of Lemma 3.1 relies on a result of Valentin Gutev, Theorem 3.1 (and Remark 3) of [14]. For our purposethe full strength of that result is not needed we therefore state it, with the necessary definitions, in a truncated form.

Following Gutev, we say that a map Φ : Y → Z is lower locally constant if, for all compact subset K of Z the set{y ∈ Y : K ⊂ Φy} is open in Y . Equivalently, Φ is lower locally constant if

⋂z∈K Φ−1z is open in Y whenever K is

a compact subset of Z. In particular, a lower locally constant map has open fibers (for all z ∈ Z, Φ−1z is open in Y ),1

and is consequently lower semicontinuous.

Theorem 3.3 (A truncated formulation of Gutev’s theorem). A lower locally constant map Φ : Y → Z with nonemptyhomotopically trivial values in a topological space Z, defined on a finite dimensional paracompact space Y , has acontinuous selection.

Proof of Lemma 3.1. We have to see that |SR(Ω)| : Y → |SR| is lower locally constant. First, we verify that SR(Ω) :Y → SR , where SR is endowed with the discrete topology, is lower locally constant. Let {A0, . . . ,Am} ⊂ SR andy� ∈ Y such that {A0, . . . ,Am} ⊂ SR(Ωy�). We have Ωy� ∩ (

⋂x∈Ai

Rx) �= ∅ for all i ∈ {0, . . . ,m}. Since Ω is lowersemicontinuous and all the

⋂x∈Ai

Rx are open we can find, for each i ∈ {0, . . . ,m}, an open neighborhood Vi of y�

such that, for all y ∈ Vi , Ωy ∩ (⋂

x∈AiRx) �= ∅. For all y ∈ V = ⋂m

i=0 Vi we have {A0, . . . ,Am} ⊂ SR(Ωy). Thisshows that SR(Ω) is lower locally constant.

We show that |SR(Ω)| : Y → |SR| is lower locally constant.Let K be a compact subset of |SR| contained in |SR(Ωy�)|, for some y� ∈ Y . There is a finite number of simplices

{σ0, . . . , σm} of |SR(Ωy�)| such that K ⊂ ⋃mi=0 σi ⊂ |SR(Ωy�)|. Let Ai ∈ SR(Ωy�) be the set of vertices of σi . By

the first part of the proof there is an open neighborhood V of y� such that, for all y ∈ V and all i ∈ {0, . . . ,m},σi ⊂ |SR(Ωy)|, and therefore, K ⊂ |SR(Ωy)|. �Proof of Theorem 3.1. Let r : Y → |SR| ⊂ |〈X〉| be a continuous selection of |SR(Ω)| obtained from Lemma 3.1.For each vertex {x} of |SR|, that is for each x ∈ X such that Ωy ∩Rx �= ∅, let αx : |SR| → [0,1] be the corresponding

1 The fibers of a Φ : Y → Z are the sets Φ−1z.


barycentric map (if σ is a simplex of |SR| for which {x} is not a vertex then αx is identically 0 on σ , αx is therefore welldefined and continuous). Let S0

R be the set of vertices of |SR| and, for {x} ∈ S0R , let rx = σx ◦ r ; it is a continuous map

from Y to [0,1] and {rx : {x} ∈ S0R} is a partition of unity on Y . For all y ∈ Y there is a unique simplex σy ⊂ |SR| such

that r(y) belongs to the interior of that simplex; the set of vertices of that simplex is ρ(y) = {x: σx(r(y)) > 0}. Themap ζ : Y → [0,1] defined by ζ(y) = sup{x}∈S0

Rrx(y) is lower semicontinuous. Let ω(y) = {x ∈ X: rx(y) � 1

2ζ(y)},it is nonempty by definition of ζ(y). Since for all y ∈ Y there exists x ∈ X such that rx(y) > 0 we have rx(y) > 0 forall x ∈ ω(y) and therefore ω(y) ⊂ ρ(y); this shows that ω(y) is finite. Also, from

∑{x}∈ρ(y) σx(r(y)){x} = r(y) ∈

|SR(Ωy)| we have ρ(y) ∈ SR(Ωy); from ω(y) ⊂ ρ(y) we also have ω(y) ∈ SR(Ωy), that is Ωy ∩ (⋂

x∈ω(y) Rx) �= ∅.

For all y ∈ Y let W 1y = {y′ ∈ Y : 1 − ∑

x∈ρ(y) rx(y′) < 1

2ζ(y′)}; from the lower semicontinuity of ζ it is an open

set and it contains y since 1 − ∑x∈ρ(y) rx(y) = 0 < 1

2ζ(y). Notice that if x /∈ ρ(y) and y′ ∈ W 1y then rx(y

′) �1 − ∑

x∈ρ(y) rx(y′) which yields rx(y

′) < 12ζ(y′), and therefore x /∈ ω(y′).

For all x ∈ ρ(y)\ω(y) choose an open neighborhood W 2x,y of y such that, for all y′ ∈ W 2

x,y , rx(y′) < 1

2ζ(y′) and let

W 2y = ⋂

x∈ρ(y)\ω(y) W2x,y ; it is an open neighborhood of y. By construction, for all x ∈ ρ(y)\ω(y) and for all y′ ∈ W 2

y

we have rx(y′) < 1

2ζ(y′). Finally, let Vy = W 1y ∩ W 2

y . We have seen that if y′ ∈ Vy and if x /∈ ω(y) then x /∈ ω(y′); inother words, for all y′ ∈ Vy we have ω(y′) ⊂ ω(y). �Proof of Theorem 3.2. Let {Rn: n ∈ N} be a countable base of U such that, for all n„ and Rn+1 ⊂ Rn and R−1

n = Rn,and, for all n and all x ∈ X, Rnx is open. For all n there exists R ∈ Uconv such that R ⊂ Rn therefore, by Theorem 3.1there is a lower-semicontinuous map Γ0 : Y → X with nonempty closed convex values and a single valued mapγ0 : Y → X such that Γ0y ⊂ Ωy ∩ R0γ0(y) for all y. Assuming that we have constructed Γn : Y → X, lower semi-continuous with nonempty closed convex values, we obtain from Theorem 3.1 Γn+1 : Y → X with nonempty closedconvex values and a single valued map γn+1 : Y → X such that Γn+1y ⊂ Γny ∩ Rn+1γn+1(y) for all y. Since theuniformity is complete there is for all y ∈ Y a point f (y) ∈ X such that

⋂n∈N

Γny = {f (y)}.By construction we have f (y) ∈ Ωy for all y ∈ Y . To see that f is continuous fix an arbitrary point y ∈ Y , let

W ⊂ X be a neighborhood of f (y), choose l such that Rl ◦ Rl(f (y)) ⊂ W and n such that Rn ◦ Rn ⊂ Rl . ThenΓny ⊂ Rl(f (y)) for all y ∈ Y . From the lower semicontinuity of Γl there exists a neighborhood V of y in Y such that,for all y ∈ V , Γny ∩ Rl(f (y)) �= ∅, consequently, Rl(f (y)) ∩ Rl(f (y)) �= ∅ for all y ∈ V and f (y) ∈ Rl ◦ Rl(f (y))

for all y ∈ V ; we have shown that f (V ) ⊂ W. �Corollary 3.1.

(1) A completely metrizable uniform convex space (X,U,C) with the Van de Vel Property is an absolute extensor forthe class of paracompact finite dimensional spaces. If X is finite dimensional then it is contractible.

(2) A completely metrizable uniform convex space (X,U,C) with the Van de Vel Property is homotopically trivial.

Corollary 3.2.

(1) A completely metrizable convex subset of a Van de Vel space is homotopically trivial.(2) Completely metrizable convex subsets of a uniform convex space (X,U,C) with property (S4-weak) and with

compact connected polytopes are homotopically trivial.

Proof. We have observed in the previous section that convex subsets of a Van de Vel space have, for the induced uni-formity and the induced convexity, the Van de Vel Property. Part (1) is therefore a consequence of (2) of Corollary 3.1.

The second part is a consequence of the first and of Proposition 2.4. �3.2. Selections for lower semicontinuous maps

We show in this section that Michael’s Selection Theorem can be adapted to uniform convex spaces, more ex-actly:


Theorem 3.4 (Michael’s Selection Theorem). Let (X,U,C) be a uniform convex space for which polytopes are ho-motopically trivial. If the uniformity is completely metrizable then all lower semicontinuous maps Ω : Y → X withnonempty closed convex values defined a paracompact space Y have a continuous selection.

If A ⊂ Y is a closed set then any continuous selection of Ω restricted to A extends to a continuous selection of Ω .

Without assuming that the uniformity is metrizable, we show that a lower semicontinuous map with nonemptyconvex values, defined on a paracompact topological space has, for all elements R of the uniformity, an R-selection,in the sense that the each value of the selection is R-close to the corresponding value of the convex valued map.

The proof of the extension of Michael’s Theorem is done through the construction of an appropriate sequence ofsuch approximate selections and by taking the pointwise limit of these approximate selections. All the preliminaryresults are presented first.

Lemma 3.2. Let (X, τ,C) be a topological space endowed with a convexity C for which polytopes are homotopicallytrivial. Assume that we are given a paracompact topological space Y , an open covering R of Y and, for all U ∈ R,a point η(U) ∈ X; for all y ∈ Y let σ(y;R) = {U ∈ R: y ∈ U}. Then, there exists a continuous map g from thegeometric realization of the nerve of R to X such that, for all y ∈ Y , g(y) ∈ �{η(U): U ∈ σ(y;R)}�.Proof. Let |R| be the geometric realization of the nerve of R; identifying the set of vertices of |R| with R, we alreadyhave a continuous map η : |R|0 → X. Assume that we have a continuous map ηm : |R|m → X such that, for all m-di-mensional simplex σ , ηm(σ ) ⊂ �{η(U): U ∈ σ 0}�. If σ is an (m + 1)-dimensional simplex then ηm restricted to theboundary of σ is a continuous map into �{η(U): U ∈ σ 0}�, and, since this set is homotopically trivial, we can extendηm restricted to the boundary of σ to a continuous map ηm,σ from σ to �{η(U): U ∈ σ 0}�; if σ and σ ′ are two (m+1)-dimensional simplices which intersect then ηm,σ and ηm,σ ′ agree on σ ∩σ ′; we therefore have a continuous map ηm+1defined on |R|m+1 such that ηm+1(σ ) ⊂ �{η(U): U ∈ σ 0}� for all (m + 1)-dimensional simplex. By induction, weobtain a continuous map η : |R| → X such that, for all simplices, η(σ ) ⊂ �{η(U): U ∈ σ 0}�.

Since Y is paracompact, there exists a locally finite partition of unity {κU : U ∈ R} on Y such that κ−1U (]0,1]) ⊂ U

for all U ∈ R; let κ : Y → |R| be the associated canonical map. For all y ∈ Y denote by σ(y) the smallest simplexof |R| to which κ(y) belongs; if U is a vertex of σ(y) the κU(y) > 0 and therefore y ∈ U , this shows that σ(y)0 ⊂σ(y;R), and therefore η(κ(y)) ⊂ �{η(U): U ∈ σ(y)0}�⊂ �{η(U): U ∈ σ(y;R)}�. We can take g = η ◦ κ . �Proposition 3.1. Let (X,U,C) be a uniform convex space for which polytopes are homotopically trivial and Ω : Y →X a lower semicontinuous map with nonempty convex values from a paracompact topological space Y to X. Then,for all R ∈ U , there exists a continuous map g : Y → X such that, for all y ∈ Y , g(y) ∈ R(Ωy).

Proof. Given R ∈ U there exists S ∈ U such that, for all C ∈ U , �S(C)� ⊂ R(C); without loss of generality, we canassume that S−1x is open for all x ∈ X. For all y ∈ Y , let Γy = S(Ωy); Γy is never empty, since Ω has nonemptyvalues, therefore {Γ −1x: x ∈ X} is a covering of Y ; for all x ∈ X, Γ −1x is open, by the lower semicontinuity of Ω

and by Γ −1x = {y ∈ Y : Ωy ∩ S−1x �= ∅}. Let V be a locally finite open covering of Y finer than {Γ −1x: x ∈ X}.For all V ∈ V there exists xV ∈ X such that V ⊂ Γ −1xV . From Lemma 3.2 there exists a continuous map g : Y → X

such that, for all y ∈ Y , g(y) ∈ �{xV : y ∈ V }�; notice that if y ∈ V then Ωy ∩ S−1xV �= ∅, and therefore xV ∈ S(Ωy);taking the convex hull, we obtain �{xV : y ∈ V }�⊂ �S(Ωy)�, and, from the choice of S, g(y) ∈ R(Ωy). �

The conclusion of Proposition 3.1 can be formulated somewhat differently; U has a base of symmetric relations,therefore, in the statement of Proposition 3.1 we can replace R by R−1 and the conclusion becomes R(g(y))∩Ωy �= ∅for all y ∈ Y.

Lemma 3.3. Let X and Y be topological spaces, Ω : Y → X a lower semicontinuous map, g : Y → X a continuousmap and R ⊂ X × X an open graph relation. Then y �→ R(g(y)) ∩ Ωy defines a lower semicontinuous map from Y

to X.

Proof. Notice that y �→ {g(y)} × Ωy is lower semicontinuous from Y to X × X, call this map Ψ . If U ⊂ X is openthen R ∩X ×U is open in X ×X and, for all y ∈ Y , R(g(y))∩Ωy ∩U �= ∅ if and only if Ψy ∩ (R ∩X ×U) �= ∅. �


Lemma 3.4. If (X,U,C) is a uniform convex space and Ω : Y → X is a lower semicontinuous map then y �→ �Ωy�is also lower semicontinuous.

Proof. Let V ⊂ X be an open set and y ∈ Y such that �Ωy� ∩ V �= ∅. There is a finite set {x0, . . . , xm} ⊂ Ωy suchthat �{x0, . . . , xm}� ∩ V �= ∅. Choose u ∈ �{x0, . . . , xm}� ∩ V and R ∈ U such that Ru ⊂ V . There exists S ∈ U suchthat, for all convex set C ⊂ X, �S(C)� ⊂ R−1(C); without loss of generality, we can assume that, for all x ∈ X, S−1x

is open.By reflexivity of S we have S−1xi ∩ Ωy �= ∅ for all i ∈ {0, . . . , n}; let Wi ⊂ Y be an open neighborhood of y such

that, for all y ∈ Wi , S−1xi ∩ Ωy �= ∅. Then for all y ∈ W = ∩ni=0Wi we have {x0, . . . , xn} ⊂ S(Ωy) and, a fortiori,

{x0, . . . , xn} ⊂ S(�Ωy�) and therefore �{x0, . . . , xn}� ⊂ �S(�Ωy�)�. From the choice of S we get, for all y ∈ W ,�{x0, . . . , xn}� ⊂ R−1(�Ωy�). Since u belongs to the left-hand side of the last inclusion we have Ru ∩ �Ωy� �= ∅ forall y ∈ W , finally, from Ru ⊂ V we get V ∩ �Ωy� �= ∅ for all y ∈ W . �Proof of Theorem 3.4. Let d : X × X → R+ be a complete metric for which the relations Rn = {(x, y): d(x, y) <

2−n} form a base of the uniformity U . Since Uconv is also a base of U there exists T0 ∈ Uconv and n0 > 0 such thatRn0 ⊂ T0 ⊂ R0. By Proposition 3.1 there is a continuous map g0 : Y → X such that, for all y ∈ Y , Ωy ∩Rn0g0(y) �= ∅,this implies Ωy ∩ R0g0(y) �= ∅. Since Ωy is convex for all y ∈ Y and T0x is convex for all x ∈ X we have, for ally ∈ Y , �Ωy ∩ Rn0g0(y)� ⊂ Ωy ∩ T0g0(y) ⊂ Ωy ∩ R0g0(y).

Let Ω1y = �Ωy ∩ Rn0g0(y)�; by Lemmas 3.3 and 3.4 Ω1 is lower semicontinuous, and by construction it hasconvex values. Choose T1 ∈ Uconv and n1 > n0 such that Rn1 ⊂ T1 ⊂ R1 and repeat the previous construction to find acontinuous map g1 : Y → X such that, for all y ∈ Y , Ω1y ∩Rn1g1(y) �= ∅ and �Ω1y ∩Rn1g1(y)� ⊂ Ω1y ∩T1g1(y) ⊂Ω1y ∩ R1g1(y).

By construction we also have Ω1y ⊂ Ωy∩R0g0(y) for all y ∈ Y , from which we can conclude that Ωy∩R0g0(y)∩R1g1(y) �= ∅. An obvious induction yields a sequence gn : Y → X of continuous maps such that, for all n ∈ N andfor all y ∈ Y , Ωy ∩ R0g0(y) ∩ · · · ∩ Rngn(y) �= ∅; from Rngn(y) ∩ Rn+1gn+1(y) �= ∅ we have d(gn(y), gn+1(y)) <

2−n + 2−n−1, in other words, the sequence (gn)n∈N is uniformly Cauchy. The completeness of the metric implies thatthere is a continuous map g : Y → X such that (gn(y))n∈N converges to g(y); from Ωy ∩ Rngn(y) �= ∅ we see thatg(y) belongs to the closure of Ωy.

The last part is proved as usual: if g : A → X is a continuous selection of Ω restricted to A then

Ωy ={

g(y) if y ∈ A,

X if y /∈ A

is lower semicontinuous with closed convex values. �3.3. Approximate selections for upper semicontinuous maps

We prove the following Cellina like result, [8]:

Theorem 3.5 (Approximate selections for usc maps). Let (X,U,C) be a uniform convex space for which polytopesare homotopically trivial, Y a paracompact topological space and Ω : Y → X an upper semicontinuous map withnonempty convex values. Then, for all R ∈ U there exists a continuous map f : Y → X such that, for all y ∈ Y ,f (y) ∈ R(Ωy).

Furthermore, if the values of Ω are convex and compact then any neighborhood Θ ⊂ Y × X of the graph of Ω

contains the graph of a continuous map f : Y → X.

First, we show that in an arbitrary uniform convex space, a uniform neighborhood of an upper semicontinuous mapwith convex values contains the graph of an open graph upper semicontinuous map with convex values; if the valuesof the initial map are also compact then all neighborhoods of its graph must contain the graph of an open graph uppersemicontinuous map with convex values. From this we will deduce in a subsequent section a Kakutani like fixed pointtheorem for compact upper semicontinuous maps (i.e. all the values are contained in a compact convex set), withclosed convex values in uniform convex spaces.


Proposition 3.2. Let (X,U,C) be a uniform convex space and Ω : Y → X an upper semicontinuous map with non-empty convex values defined on a paracompact topological space Y . Then, for all R ∈ U there exist a locally finiteopen covering {Wy}y∈Y of Y , and an open graph map Γ : Y → X with convex values such that

(1) for all y ∈ Y , y ∈ Wy and(2) for all y ∈ Y , Ωy ⊂ Γy ⊂ Γ (Wy) ⊂ R(Ωy).

Furthermore, if the values of Ω are compact and convex then, for all open neighborhood Θ ⊂ Y × X of the graphof Ω , there exists an open graph and convex valued map Γ : Y → X such that Ω ⊂ Γ ⊂ Θ.

Proof. By Proposition 2.2 there exists, for all y ∈ Y , an open convex set Vy ⊂ X such that Ωy ⊂ Vy ⊂ R(Ωy). Byassumption, Ω is upper semicontinuous, let By ⊂ Y be an open neighborhood of y such that Ω(By) ⊂ Vy and, byregularity of Y , let Oy be an open neighborhood of y such that Oy ⊂ Oy ⊂ By , and by paracompactness, one can findan open neighborhood Wy of y, contained in Oy such that {Wy}y∈Y is a locally finite covering of Y . For all y ∈ Y , theset {z ∈ Y : y ∈ Wz} is finite; let Γy = ⋂

y∈WzVz; it is an open convex set. By construction, if y′ ∈ Wy then Γy′ ⊂ Vy

therefore Γ (Wy) ⊂ R(Ωy). Also, if y ∈ Wz then y ∈ Bz, and consequently, Ωy ⊂ Vz; this shows that Ωy ⊂ Γy.To complete this first part of the proof we show that the graph of Γ is open. For all y ∈ Y let Ny = Y \⋃

y /∈WzWz;

it is an open neighborhood of the point y, therefore Ny × Γy is open in Y × X and, obviously, {y} × Γy ⊂ Ny × Γy.If (y′, x′) ∈ Ny × Γy then, for all z ∈ Y , y′ ∈ Wz implies y ∈ Wz; in other words, for all y′ ∈ Ny , Γy ⊂ Γy′, andtherefore x′ ∈ Γy′. This shows that (y′, x′) ∈ Γ , and consequently, that, for all y ∈ Y , Ny × Γy ⊂ Γ . We have shownthat Γ is an open subset of Y × X.

To establish the last part of the proposition, notice first that for all compact set K ⊂ X and for all open set U ⊂ X

containing K there exists R ∈ C such that K ⊂ R(K) ⊂ U . Since Θy is an open set containing Ωy there exists, byCorollary 2.1, an open convex set Vy such that Ωy ⊂ Vy ⊂ Θy; we can now proceed as in the first part of the proof,with Θy replacing R(Ωy). �

Proposition 3.2 and its proof are slight adaptations to our framework of Ancel’s Enlargement Lemma in [1].

Lemma 3.5. If (X,U,C) is a uniform convex space for which polytopes are homotopically trivial then any mapΓ : Y → X with nonempty convex values and open graph, from a paracompact topological space Y to X, has acontinuous selection.

Proof. Notice that {Γ −1x: x ∈ X} is an open covering of Y and, taking into account that the values Γy are convex,we can proceed as in the proof of Proposition 3.1. �Proof of Theorem 3.5. From Proposition 3.2 there exists a convex valued map with open graph Γ : Y → X suchthat, in the first case, Γy ⊂ R(Ωy) for all y ∈ Y , in the second case, Γ ⊂ Θ . An appeal to Lemma 3.5 completes theproof. �4. Dugundji’s theorem in uniform convex spaces

As the title of this section indicates we establish Theorem 4.1 below. All the necessary ingredients to closely followDugundji’s proof in [12] are at our disposal.

Theorem 4.1. A uniform convex space for which polytopes are homotopically trivial is an absolute extensor for theclass of metric spaces.

Proof. Let (Y, d) be a metric space and f : A → X a continuous map defined on a closed subset A of Y . For ally ∈ Y \ A choose an open ball By centered at y of radius strictly smaller than 2−1d(y,A), the distance from y to A

and let R be a locally finite open covering of Y \ A finer than {By : y ∈ Y \ A}. To each U ∈ R associate a pair ofpoints (aU , xU ) ∈ A × U such that d(aU , xU ) < 2d(xU ,A).


By Lemma 3.2, there exists a continuous map h : Y \ A → X such that, for all y ∈ A, h(y) ∈ �{f (aU ): U ∈σ(y;R)}�. Let

g(y) ={

f (y) if y ∈ A,

h(y) if y ∈ Y \ A.

Since g is continuous on the open set Y \ A we only have to check the continuity at all point of A to establish that g

is continuous on Y ; this is accomplished as in [12]. First, for all a ∈ A and for all neighborhood W(a) of a in X thereexists a neighborhood V (a) of a in X such that V (a) ⊂ W(a) and, for all U ∈ R, if U ∩ V (a) �= ∅ then U ⊂ W(a)

and aU ∈ W(a).Let a ∈ A and let W ⊂ X be a neighborhood of g(a); there is a convex neighborhood C ⊂ W of g(a) = f (a)

and a neighborhood W(a) of a in X such that f (A ∩ W(a)) ⊂ C. Let V (a) be as above, then f (A ∩ V (a)) ⊂ C; lety ∈ Y \ A, if U ∈ R and y ∈ U ∩ V (a) then aU ∈ A ∩ W(a) and therefore f (aU ) ∈ C, this shows that {f (aU ): U ∈σ(y;R)} ⊂ C, and, since C is convex, that h(y) ∈ C. We have shown that h(V (a) ∩ Y \ A) ⊂ C which, together withf (A ∩ V (a)) ⊂ C, gives us g(V (a)) ⊂ C. �

The main, and immediate, consequence of Theorem 4.1 is the following corollary:

Corollary 4.1. A metrizable uniform convex space with homotopically trivial polytopes is an absolute retract.

5. Fixed points

Brouwer’s Fixed Point Theorem can be derived from the well known result of Knaster–Kuratowski–Mazurkiewiczknown as the KKM Lemma, [13]; in this section we investigate convexities for which a KKM like property holds. InSection 5.1 we first recall how to prove, in an almost formal way, fundamental intersection and fixed point theoremsfor multivalued maps if we have a KKM like property at our disposal. Then, under an additional hypothesis (which inthe classical linear setting is verified if there are enough continuous linear maps on compact convex sets to separatepoints), we derive a Schauder–Tychonov like Fixed Point Theorem. We establish the KKM Property for uniformconvex spaces with the Van de Vel Property and also for uniform convex spaces with compact connected polytopeswith the Kakutani Property.

In the second part, Section 5.2, we prove the Kakutani–Fan–Himmelberg Fixed Point Theorem for upper semicon-tinuous maps, Theorem 5.2, and therefore, the Schauder–Tychonov Fixed Point Theorem, Corollary 5.2, in uniformconvex spaces with the Kakutani Property and with compact, connected metrizable polytopes.

Invoking Corollary 4.1 we can immediately state that a metrizable uniform convex space with homotopicallytrivial polytopes has the fixed point property for continuous compact maps (recall that a continuous map f : X → X

is compact if the closure of its image is a compact subset of X). But neither the metrizability condition nor theuniform continuity of the convex hull operator are a priori necessary conditions for fixed point properties, as shown,for example, by Proposition 5.4.

5.1. Around KKM

The results of this section lay down a formal framework for fixed point properties. Accordingly, most of the proofs,maybe with the exception of that of Theorem 5.1, cannot claim much originality.

Definition 5.1. Let X be a topological space endowed with a convexity C; we say that (X,C) has the weak Van de VelProperty if polytopes are compact and are, with the induced convexity, uniform convex spaces with the Van de VelProperty.

Convex subsets of arbitrary topological vector spaces have the weak Van de Vel Property.The second condition is unambiguous since a compact space has a unique uniformity compatible with its topology.

Corollary 2.3 implies that arbitrary topological vector spaces have the weak Van de Vel Property.If (X,C) has the weak Van de Vel Property then, by Lemma 2.2, polytopes are also connected. On the other hand,

if polytopes are compact and connected and if the Kakutani separation property holds for convex subsets of polytopes


(or more generally the S4-weak property) with the induced convexity then (X,C) has the weak Van de Vel Propertyholds.

Definition 5.2. An indexed family of subsets (A0, . . . ,Am) of a set X is a KKM family with respect to a convexityC on X if there exists a finite set of points {p0, . . . , pm} ⊂ X such that for all set of indices J ⊂ {0, . . . ,m} we have�{pj : j ∈ J }�⊂ ⋃

j∈J Aj .The convexity C is KKM with respect to the family A of subsets of X if for all indexed family of subsets (A0,

. . . ,Am) composed of elements of A which is KKM with respect to C we have⋂m

i=0 Ai �= ∅.

One translation of the classical KKM Lemma in this language is that the usual convexity of a convex subset of anarbitrary topological vector space is KKM with respect to the family of closed sets.

From Theorem 1 in [17] a convexity with homotopically trivial polytopes is KKM with respect to the family ofclosed sets and with respect to the family of open sets; recalling Corollary 3.1 we obtain

Proposition 5.1. If (X,C) has the weak Van de Vel Property and if polytopes are metrizable then C is KKM withrespect to the family of closed subsets of X and also with respect to the family of open subsets of X.

Without the metrizability assumption polytopes might not be homotopically trivial, and therefore Theorem 1 in[17] is not available anymore. But the KKM property still holds for closed sets.

Theorem 5.1. If (X,C) has the weak Van de Vel Property then C is KKM with respect to the family of closed sets.

Proof. First, we notice that we can assume that X itself is compact and that C is compatible with the uniformity ofX; indeed, if P = �{p0, . . . , pm}� is a polytope such that �{pj : j ∈ J }� ⊂ ⋃

j∈J Fj , where (F0, . . . ,Fm) is a familyof closed subsets of X, we can replace X by P and Fj by P ∩ Fj .

(A) Next, we claim (and we will prove it in (C)), that if (X,U,C) is a uniform convex space with the Van de VelProperty and if (A0, . . . ,Am) is a family of open sets which is KKM with respect to C then, for all R ∈ U there existsa lower semicontinuous map ΓR : Δm → X, where Δm is the standard m-dimensional simplex, with closed nonemptyconvex values and a single valued map γR : Δm → X such that:

(1) for all set of indices J , ΓR(ΔJ ) ⊂ ⋃j∈J Aj and

(2) for all y ∈ Δm, ΓRy ⊂ RγR(y).

Assuming that (A) holds, let ωR,j = {y ∈ Δm: ΓRy ∩ Aj �= ∅}; from the lower semicontinuity of ΓR we have thatωR,j is an open subset of Δm and from (2) we have ΔJ ⊂ ⋃

j∈J ωR,j for all sets of indices J ⊂ {0, . . . ,m}. Fromthe standard KKM Theorem, for open sets, we have

⋂mj=0 ωR,j �= ∅; if y ∈ Δm is a point of

⋂mj=0 ωR,j then, by (2),

RγR(y) ∩ Aj �= ∅ for all j . We have shown that

(3) for all R ∈ U ,⋂m

j=0 R−1Aj �= ∅.

(B) Now we assume that (X,U,C) is a compact (uniform) convex space with the Van de Vel Property and that(F0, . . . ,Fm) is a family of closed sets which is KKM with respect to C. For all R ∈ U choose S ∈ U such that S = S−1

and S ◦S ⊂ R; then SA ⊂ RA for all subsets A ⊂ X. From (3) applied to S we obtain⋂m

j=0 RAj �= ∅. If R ⊂ R1 ∩R2

then⋂m

j=0 RAj ⊂ (⋂m

j=0 R1Aj) ∩ (⋂m

j=0 R2Aj); since X is compact we must have⋂

R∈U (⋂m

j=0 RAj) �= ∅ and this

implies that⋂m

j=0 Aj �= ∅.(C) To complete the proof we establish the claim made in A. For all y = (t0, . . . , tn) ∈ Δn let σ(y) = {i: ti >

0}; given {p0, . . . , pn} ⊂ X such that �{pj : j ∈ J }� ⊂ ⋃j∈J Aj for all nonempty subset of indices J ⊂ {0, . . . , n},

let Ωy = �{pj : j ∈ σ(y)}�. By definition, Ω : Δn → �{p0, . . . , pn}� has convex values; we show that it is lowersemicontinuous. Let V (y) = {y′ ∈ Δn: σ(y) ⊂ σ(y′)}; by continuity of the barycentric coordinate maps V (y) isan open neighborhood of y, furthermore, for all y′ ∈ V (y) we have Ω(y) ⊂ Ω(y′) and consequently, for all U ⊂�{p0, . . . , pn}�, if Ωy ∩ U �= ∅ then Ωy′ ∩ U �= ∅ for all y′ ∈ V (y). The claim now follows from Theorem 3.1. �


Corollary 5.1. A convexity C on a topological space X for which

(1) polytopes are compact and connected;(2) for all polytopes, the induced convexity is compatible with the uniformity;(3) all polytopes satisfy S4-weak

is KKM with respect to the family of closed sets. Furthermore, if polytopes are also metrizable then C is also KKMwith respect to the family of open sets.

Given a topological space X endowed with a convexity C we say that a map Ω : Y → X is KKM with respect tothe convexity C, or simply KKM, if, for all nonempty finite subset {y0, . . . , ym} ⊂ Y the family {Ωy0, . . . ,Ωym} isKKM. From the definitions we have the well known KKM principle.

Proposition 5.2. If C is a convexity on the topological space which is KKM with respect to the family of closed setsand if Ω : Y → X is a KKM map with closed values, at least one of which is compact, then

⋂y∈Y Ωy �= ∅.

If Y is a subset of X we say that Ω : Y → X is a Ky Fan map, with respect to the convexity C on X, if:

(KF1) for all y ∈ Y , y ∈ Ωy;(KF2) for all x ∈ X, Y \ Ω−1x is convex.

Lemma 5.1. Ky Fan maps are KKM maps.

Proof. We show that for all finite subsets {y0, . . . , ym} of Y , �{y0, . . . , ym}�⊂ ⋃ni=0 Ωyi . For a contradiction, assume

that there exists a finite nonempty subset {y0, . . . , ym} of Y such that �{y0, . . . , ym}� is not contained in⋃n

i=0 Ωyi ;there exists z ∈ �{y0, . . . , ym}� such that z /∈ ⋃n

i=0 Ωyi , this is equivalent to {y0, . . . , ym} ⊂ Y \ Ω−1z, and therefore,by the convexity assumption, we obtain �{y0, . . . , ym}� ⊂ Y \ Ω−1z, which implies z /∈ Ωz, in contradiction with(KF1). �

Proposition 5.2 and Lemma 5.1 yield the following proposition:

Proposition 5.3 (KKM principle). If (X,C) is a topological space which is endowed with a convexity which is KKMwith respect to closed sets and if Y is a subset of X then, for all Ky Fan maps with closed values Ω : Y → X, at leastone of which is compact, we have

⋂y∈Y Ωy �= ∅.

The most immediate fixed point result is:

Proposition 5.4 (Fan–Browder’s Fixed Point Theorem). Let (X,C) be a topological space which is endowed with aconvexity which is KKM with respect to closed sets and let Y be a compact convex subset of X. If Γ : X → X is a mapwith nonempty convex values such that, for all y ∈ Y , Γ −1y is open and Γy ∩ Y �= ∅ then there exists y� ∈ Y suchthat y� ∈ Γy�.

Proof. For a contradiction, assume that Γ : X → X has no fixed point; then for all x ∈ X, x ∈ X \ Γ −1x. LetΩx = X \ Γ −1x; since X \ Ω−1x = Γ x for all x ∈ X, we have that Ω : X → X is a Ky Fan map with closed values.For all y ∈ Y , y ∈ Ωy ∩ Y , and since Y is convex, y �→ Ωy ∩ Y is a Ky Fan map with compact values and therefore⋂

y∈Y Ωy ∩ Y �= ∅, or Y ∩ (X \ ⋃y∈Y Γ −1y) �= ∅, in other words, there exists y� ∈ Y such that Γy� ∩ Y = ∅, which

is a contradiction. �Definition 5.3. A convexity C on a topological space X is a Φ-convexity if for all compact convex subset Y of X andfor all open set U ⊂ Y ×Y containing ΔY = {(y, y) ∈ Y ×Y } there exists R : X → X reflexive with nonempty convexvalues and open fibers such that R ∩ (Y × Y) ⊂ U . If C is a Φ-convexity on X we will say that (X,C) is Φ-convex.


Let X and Y be to sets endowed with convexities CX and CY ; following Van de Vel, we will say that a mapf : X → Y is convexity preserving if, for all C ∈ CY the set f −1(C) belongs to CX . If for all pair of distinct points(x, x′) of X there is a continuous convexity preserving map f : X → R such that f (x) �= f (x′) we say that (X,CX)

has enough convexity preserving maps.

Lemma 5.2. A convexity C on a topological space X for which there are enough convexity preserving maps is aΦ-convexity.

Proof. Let Y be a compact convex subset of X and U ⊂ Y × Y an open set such that ΔY ⊂ U . For all pairs (x, x′) ∈Y ×Y \U there exists a continuous convexity preserving map ϕ(x,x′) : X → R such that ϕ(x,x′)(x)−ϕ(x,x′)(x′) > 0; de-fine f(x,x′) : X ×X → R by f(x,x′)(y, y′) = ϕ(x,x′)(y)−ϕ(x,x′)(y′) and choose λ(x,x′) in ]0, f(x,x′)(x, x′)[. From com-pactness of Y ×Y \U there is a finite set {(x1, x

′1), . . . , (xm, x

′m)} such that Y ×Y \U ⊂ ⋃m

i=1{(x, x′): f(xi ,x

′i )(x, x′) >

λ(xi ,x

′i )}; choose λ in the interval ]0,min1�i�m{λ

(xi ,x′i )}[ and let f (x, x′) = max1�i�m{f

(xi ,x′i )(x, x′)}. Notice that f

is continuous, that for all x ∈ X, f (x, x) = 0 and therefore Y × Y \ U ⊂ {(x, x′): λ � f (x, x′)}.The relation R = {(x, x′) ∈ X × X′: f (x, x′) < λ} is reflexive, since 0 < λ; also Rx = {x′ ∈ X: f (x, x′) < λ}

= ⋂mi=1{x′ ∈ X: f

(xi ,x′i )(x, x′) < λ} is convex and R−1x′ = {x ∈ X: f (x, x′) < λ} is open; and we have ΔY ⊂

(Y × Y ∩ R) ⊂ U by construction. �Lemma 5.3. Let (X,C) be Φ-convex and assume that C is KKM with respect to closed sets; then, any continuous mapf : X → Y from X to a compact convex subset Y of X has a fixed point.

Proof. Let R : X → X be a reflexive map with nonempty convex values and open inverse images and let Γ x = Rf (x).Since f : X → X is continuous and Γ −1x = f −1(R−1(x)) we have that Γ has nonempty convex values and Γ −1x isopen for all x ∈ X; furthermore, for y ∈ Y we have f (y) ∈ Y ∩Rf (y), by the reflexivity of R. Both ΔY and GY (f ) ={(y, f (y)): y ∈ Y } are closed subsets of the compact set Y × Y ; if f (y) �= y for all y ∈ Y then ΔY ∩ GY (f ) = ∅. Bycompactness, there is an open set U ⊂ Y × Y containing ΔY such that U ∩ GY (f ) = ∅; the convexity being Ky Fanthere exists R such that ΔY ⊂ (R ∩ Y × Y) ⊂ U , and therefore (R ∩ Y × Y) ∩ GY (f ) = ∅, contrary to what has beenestablished at the beginning of the proof. �

The next statement is an obvious consequence of Lemmas 5.2 and 5.3.

Lemma 5.4. If there is on a compact topological space X a convexity C which is KKM with respect to closed sets andfor which there are enough convexity preserving maps, then any continuous map f : X → X has a fixed point.

5.2. Fixed points

We begin with an intermediary result which surprisingly links the fixed point property to separation properties. In itsfinal version, without the completeness assumption on the uniformity, that result will simply be Schauder–Tychonov’sFixed Point Theorem in uniform convex spaces.

Proposition 5.5. Let (X,U,C) be a complete uniform convex space with compact and connected polytopes. If one ofthe following sets of hypotheses holds then all continuous compact maps f : X → X have a fixed point.

(A) all polytopes satisfy S4-weak and there are enough convexity preserving maps or(B) the convexity has the Kakutani Property.

Proof. According to Theorem 2.8 in [26] the closed convex hull of a compact set is compact. If (A) holds then theconclusion follows from Lemma 5.4 and Corollary 5.1.

Assume that (B) holds. Then, according to Theorem 2.7 in [26] there are enough continuous convexity preservingmaps from X to R, and therefore (A) is also the case. �


Lemma 5.5. Let (X,U) be a Hausdorff uniform space and Ω : X → X an upper semicontinuous compact map withclosed values, then Ω has a fixed point if and only if, for all R ∈ U , there exists xR ∈ X such that xR ∈ R(ΩxR).

Proof. Let K be the closure of Ω(X); for all R ∈ U there exists xR ∈ X and uR ∈ ΩxR such that xR ∈ RuR ; (uR)R∈Uis a generalized sequence in K . By compactness of K , there exists a base B of U such that the generalized sequence(uR)R∈B converges in K ; let x be its limit. Let T ∈ U be an arbitrary element of the uniformity and choose S ∈ Usuch that S ◦ S ⊂ T . There exists R0 ∈ B such that, for all R ∈ B with R ⊂ R0, uR ∈ Sx; without loss of generalitywe can assume that R0 ⊂ S, from which we obtain RuR ⊂ S(Sx) if R ⊂ R0; since xR ∈ RuR for all R, we haveshown that for all R ∈ B with R ⊂ R0, xR ∈ T x. Since T was an arbitrary element of U the generalized sequence(xR)R∈B converges in X to x. If x /∈ Ωx then there exists R ∈ U such that x /∈ R(Ωx) (since, for all A ⊂ X,A =⋂

R∈U R(A)); choose R1 ∈ U , symmetrical, such that R1 ◦R1 ⊂ R, then R1x ∩R1(Ωx) = ∅. By upper semicontinuityof Ω there is a neighborhood V of x such that ΩV ⊂ R1(Ωx); we can also find R0 ∈ B such that, for all R ∈ Bwith R ⊂ R0, xR and uR are in V , but then uR ∈ R1(Ωx). If V ⊂ R1x, which we can always assume, we obtaina contradiction. �Lemma 5.6. Let (X,U,C) be a uniform convex space such that convex hulls of compact sets are compact. If Γ : Y →X is an upper semicontinuous map with compact values then y �→ �Γy� is upper semicontinuous.

Proof. Let U ⊂ X be an open set and y a point in Y such that �Γ y� ⊂ U . By Corollary 2.1 there exists an openconvex set V ⊂ X such that �Γ y� ⊂ V ⊂ U . From Γ y ⊂ �Γ y� and the upper semicontinuity of Γ there exists anopen neighborhood Wy of y such that Γ (Wy) ⊂ V , finally, by convexity of V , we have �Γy�⊂ V for all y ∈ Wy . �Theorem 5.2 (Fan–Himmelberg’s theorem). A uniform convex space (X,U,C) with compact, metrizable and con-nected polytopes with the Kakutani Property has the fixed point property for upper semicontinuous compact mapswith closed convex nonempty values.

Proof. First, we assume that there exists a compact convex set K such that Ω(X) ⊂ K . Consider K with the induceduniformity U |K and the induced convexity C|K ; (K,U |K,C|K) is a uniform convex space with compact and connectedpolytopes, and with the Kakutani Property. The map x �→ Ωx ∩ K = Ωx is upper semicontinuous with nonemptyconvex and compact values. Let ΔK = {(x, x): x ∈ K}; if Ω has no fixed point then (K × K) \ ΔK is an openneighborhood of the graph of Ω : K → K . Being compact, K is paracompact, there is therefore, by Theorem 3.5, acontinuous map f : K → K whose graph is contained in (K × K) \ ΔK .

By compactness, the induced uniformity C|K is complete, and by part (B) of Proposition 5.5, there is a point x ∈ K

such that f (x) = x. We have reached a contradiction.Next, we remove the convexity assumption on K . That is K ⊂ X is compact and Ω(X) ⊂ K .Since all uniformities have a base of symmetric closed graph relations, given R ∈ U we can find S ∈ U , symmetric

and closed, such that, for all convex sets C, �S(C)� ⊂ R(C). Since K is compact we can find a finite subset F ⊂ K

such that K ⊂ S(F ); let P = �F �, we have K ⊂ S(P ). For x ∈ P we have Ωx ⊂ K and therefore Ωx ⊂ S(P ) and,since S is symmetric S(Ωx) ∩ P �= ∅. From the compactness of the values of Ω and the closedness of S we have thatS(Ωx) is closed for all x and that x �→ S(Ωx) ∩ P is an upper semicontinuous map, with compact values, from P

to P . Since P is compact, the induced uniformity is complete, and therefore the convex hull of a compact set in P iscompact (Theorem 2.8 in [26]); from Lemma 5.6 we infer that x �→ �S(Ωx)∩P � is an upper semicontinuous map withcompact convex values from P to P . From the first part of the proof there exists x ∈ P such that x ∈ �S(Ωx) ∩ P �;from �S(Ωx) ∩ P � ⊂ �S(Ωx)� ⊂ R(Ωx) we have x ∈ R(Ωx). Lemma 5.5 completes the proof. �Corollary 5.2 (Schauder–Tychonov’s Theorem). A uniform convex space (X,U,C) with compact, metrizable andconnected polytopes with the Kakutani Property has the fixed point property for continuous compact maps.


Appendix A. On the Kakutani Property

Definition A.1. A convex space (X,C) is join-hull commutative if, for all subsets S of X and for all point p ∈ X wehave �

S ∪ {p}�=⋃

x∈�S��x,p�.

Definition A.2. A convex space (X,C) has the Pash–Peano Property if for all quintuple (a, b1, b2, c1, c2) of points ofX such that ci ∈ �a, bi� one has

�b1, c2�∩ �b2, c1� �= ∅.

Proposition A.1. A join-hull commutative convex space (X,C) has the Pash–Peano Property if and only if it has theKakutani Property.

Proof. Assume that the Pash–Peano Property holds. Let Z be the family of pairs of disjoint convex sets (D1,D2)

such that Ci ⊂ Di partially ordered by (D1,D2) ⊂ (D′1,D

′2) if Di ⊂ D′

i . The pair (C1,C2) belongs to Z and ifC = {(D1,λ,D2,λ): λ ∈ Λ} is a chain in Z , that is, a totally ordered subset of Z , then (

⋃λ∈Λ D1,λ,

⋃λ∈Λ D2,λ) ∈ Z

since an up-directed union of convex sets is convex and, as can easily be seen, (⋃

λ∈Λ D1,λ) ∩ (⋃

λ∈Λ D2,λ) = ∅; byZorn’s Lemma there is a maximal element (H1,H2) in Z . Assume that there is a point a in X \(H1 ∪ H2); fromthe maximality of the pair (H1,H2) we have �H1 ∪ {a}� ∩ H2 �= ∅ and �H2 ∪ {a}� ∩ H1 �= ∅, take a point c1 in thefirst set and a point c2 in the second set. By join-hull commutativity there exists bi ∈ Hi such that ci ∈ �a, bi�. Bythe Pash–Peano Property there exists a point u in �b1, c2� ∩ �b2, c1�. From b1, c2 ∈ H1 and b2, c1 ∈ H2 we obtainu ∈ H1 ∩ H2, which is impossible since the pair (H1,H2) is in Z .

Assume now that the Kakutani Property holds but not the Pash–Peano Property. Let (a, b1, b2, c1, c2) be a quintupleof points of X such that ci ∈ �a, bi� and �b1, c2�∩ �b2, c1� = ∅. There exists a convex set C such that X \C is convex,�b2, c1� ⊂ C and �b1, c2� ⊂ X \C. We can assume that a ∈ C; then c2 ∈ �a, b2� ⊂ C. But c2 ∈ �b1, c2� ⊂ X \C; wehave reached a contradiction. �

The first part of the proof is exactly the proof of the standard algebraic Hahn–Banach Theorem in vector spaces asit can be found, for example, in the book of Holmes [15]. Proposition A.1 has no claim to originality, the proof can befound, for example, in [9], Proposition 22.

If the convexity is not joint-hull commutative then the Kakutani Property is equivalent to a generalized Pash–PeanoProperty.

Definition A.3. A convex space (X,C) has the Generalized Pash–Peano Property if for all quintuple (a,B1,B2, c1, c2),where a, c1, c2 are points of X and B1,B2 are nonempty finite subsets of X such that ci ∈ �a,Bi� one has�{c2} ∪ B1

�∩ �{c1} ∪ B2� �= ∅.

Proposition A.2. A convex space (X,C) has the Kakutani Property if and only if it has the Generalized Pash–PeanoProperty.

Proof. Assume, for a contradiction, that (X,C) has the Kakutani Property and does not have the Generalized Pash–Peano Property. We can then find (a,B1,B2, c1, c2) as above and a convex set C such that �{c2} ∪ B1� ⊂ C and�{c1} ∪ B2� ⊂ X \C. We can assume that a ∈ C; from B1 ⊂ �{c2} ∪ B1� ⊂ C and from the convexity of C we obtain�{a} ∪ B1�⊂ C. From c1 ∈ �{a} ∪ B1� we finally have c1 ∈ C ∩ �{c1} ∪ B2�⊂ C ∩ X \C.

To see that the Generalized Pash–Peano Property implies the Kakutani Property one proceeds exactly as in theproof of Proposition A.1 with the final argument modified as follows. We have two points c1 and c2 such that c1 ∈�{a} ∪ H1� ∩ H2 and c2 ∈ �{a} ∪ H2� ∩ H1. From �{a} ∪ Hi� = ⋃

B∈〈{a}∪Hi 〉�B� we find two finite sets B1 ⊂ H1 andB2 ⊂ H2 such that c1 ∈ �{a} ∪ B1� ∩ H2 and c2 ∈ �{a} ∪ B2� ∩ H1. From the Generalized Pash–Peano Property wehave �{c1} ∪ B2� ∩ �{c2} ∪ B1� �= ∅. Finally, from c1 ∈ H2, B2 ⊂ H2, c2 ∈ H1, B1 ⊂ H1 and the convexity of H1 andH2 we have �{c1} ∪ B2�∩ �{c2} ∪ B1�⊂ H1 ∩ H2, which is in contradiction with H1 ∩ H2 = ∅. �


Notice that the Generalized Pash–Peano Property holds if and only if it holds for all polytopes with the inducedconvexity. As a consequence of Proposition A.2 we obtain the following result of Keimel and Wieczorek [18].

Corollary A.1 (Keimel–Wieczorek). A convex space (X,C) has the Kakutani Property if and only if all polytopes have,with respect to the induced convexity, the Kakutani Property.

Property S4-weak is related to the screening property introduced by De Groot and Aarts in [11] where two subsetsA and B of a set X are said to be screened by a pair (C,D) of subsets of X if X = C ∪ D, A∩ C = ∅ and B ∩ D = ∅.

Condition S4-weak says that every pair of disjoint open convex subsets of a polytope is screened by a pair of closedconvex subsets of that polytope.

Appendix B. Some examples

B.1. Hyperconvex metric spaces

A metric space (X,d) is an hyperconvex metric space if for all collections {(xi, ri)}i∈I ⊂ X × R+ such that for all(i, j) ∈ I × I d(xi, xj ) � ri + rj one has

⋂i∈I B[xi, ri] �= ∅, where B[x, r] is the closed ball of radius r centered at x.

An hyperconvex metric space is complete and is a nonexpansive retract of any metric space in which it is embed-ded [2]. Fixed point and selection properties for hyperconvex metric spaces have been studied in [3,16,23] and KKMlike properties in [19].

Let C be the collection of subsets C of X such that for all nonempty finite subset A of C one has⋂{

B[x, r]: A ⊂ B[x, r]} ⊂ C.

This defines a convexity on X, the ball convexity, for which polytopes are exactly sets of the form⋂{B[x, r]:

A ⊂ B[x, r]} with A finite. With the induced metric this set is itself hyperconvex. Furthermore, for all C ∈ C and forall ε > 0

⋃

x∈C

B(x, ε) ∈ C.

This shows that the ball convexity is compatible with the metric uniformity. In conclusion:

An hyperconvex metric space equipped with the ball convexity is a uniform convex space with homotopically trivialpolytopes.

B.2. Normally supercompact spaces

A topological space (X, τ) is normally supercompact if there exists a subbase of closed sets B which is a normalfamily and such that any subfamily A ⊂ B whose members have nonempty pairwise intersections has a nonemptyintersection. Normally supercompact spaces have been extensively studied. The information needed here can be foundin one of the first paper on the subject [25]; one could also look at [16]. Given a normally supercompact topologicalspace and a closed subbase B as above let, for all finite subsets A of X,

�A�=⋂

{B ∈ B: A ⊂ B}and let C be the family of subsets C of X such that, for all finite subsets A ⊂ C, �A�⊂ C. This defines a convexity onX whose polytopes are exactly the sets �A�; since they are closed by construction they are compact. If the topology ismetrizable then there exists a metric d on X for which ε-neighborhood of polytopes are in C; also, if X is connectedthen polytopes are contractible. In conclusion:

A normally supercompact connected and metrizable topological space can be endowed with a convexity for whichit becomes a uniform convex space with homotopically trivial polytopes.


B.3. Topological semilattices

A Topological semilattice is a pair (X,�) where X is a topological space, � is a partial order on X for which eachpair of elements (x1, x2) ∈ X2 has a least upper bound x1 ∨ x2 and the map (x1, x2) �→ x1 ∨ x2 is continuous. Forfinite nonempty subsets A of X let

�A�=⋃

a∈A

[a,

∨A

]

where, for a � b, [a, b] = {x ∈ X: a � x � b} and∨

A is the least upper bound of the nonempty finite set A. DeclareC ⊂ X convex if for all nonempty finite subsets A of C one has �A� ⊂ C. This is a convexity on X whose nonemptypolytopes are the sets �A�. One can check that C is convex if and only if, for all x1, x2 ∈ C, [x1, x1 ∨ x2] ⊂ C holdsand that the Pash–Peano Property holds. If X is compact then this convexity is compatible with the uniformity of X

if each point has a neighborhood base consisting of subsemilattices and (x1, x2) �→ [x1, x1 ∨ x2] is continuous mapfrom X2 to the space of nonempty compact subspaces of X, [26], in which case, by Theorem 2.7 of [26] there areenough convexity preserving maps to separate points of X.

A lemma of Brown [7] implies that a pathconnected bounded topological semilattice is homotopically trivial;consequently if, for all x1, x2 ∈ X such that x1 � x2 the interval [x1, x1 ∨ x2] is path connected, polytopes will be pathconnected bounded semilattices and therefore homotopically trivial. Brown’s Lemma does not assume compactness.

B.4. B-convexity

In [5] and [6] a subset C of Rn+ is called B-convex if for all m ∈ N\{0}, for all (x1, . . . , xm) ∈ Cm and for all

(t1, . . . , tm) ∈ [0,1]m such that max{t1, . . . , tm} = 1 one has∨m

i=1 tixi ∈ C, where

m∨

i=1

yi =(

max1�i�m

{yi,1}, . . . , max1�i�m

{yi,n})

is the least upper bound of the vectors (y1, . . . , ym) with respect to the partial order of Rn defined by the standard

positive cone Rn+. One can check that C ⊂ R

n+ is B-convex if and only if, for all x, y ∈ C and for all t ∈ [0,1], tx ∨y ∈C. This is a convexity on R

n+ for which the Pash–Peano Property holds. Polytopes are compact and contractible. Withrespect to the metric d(x, y) = ‖x − y‖∞ = max1�j�n |xj − yj | ε-neighborhoods of B-convex sets are B-convex.

B-convexity is a uniform join-hull commutative convexity on Rn+ with the Pash–Peano Property and with compact

and connected polytopes.

For B-convexity one can directly and easily show that convex sets are contractible.

B.5. Max-Plus convexity

A subset C of (R ∪ {−∞})n is Max-Plus convex if, for all m ∈ N\{0}, for all (x1, . . . , xm) ∈ Cm and for all(t1, . . . , tm) ∈ [−∞,0]m such that max{t1, . . . , tm} = 0 one has

(max

1�i�m{xi,1 + ti}, . . . , max

1�i�m{xi,n + ti}

)∈ C.

One defines a metric on (R ∪ {−∞})n by DMax(x, y) = max1�i�n |exi − eyi |.The map x �→ (ex1 , . . . , exn) is a distance preserving homeomorphism from (R ∪ {−∞})n with the DMax metric

to Rn+ with the metric d(x, y) = ‖x − y‖∞. Max-Plus convex sets are mapped to B-convex sets and the inverse map,

that is x �→ (lnx1, . . . , lnxn), maps B-convex sets to Max-Plus convex sets.The literature on Max-Plus convexity and its applications is very large. Michael’s Selection Theorem for Max-Plus

convex sets is proved by M. Zarichnyi in [28], where additional references on Max-Plus convexity can also be found.


Appendix C. Conclusion and acknowledgement

Convexity is multifaceted; its study, and therefore its diverse generalizations, could favor the algebraic, the combi-natorial, the topological, the functional or the geometric aspect depending on one’s motivations.

A convexity like structure on a topological space might yield fixed point properties, selection theorems or ANR orAR properties. In this context the important objects are the “convex sets” or, more precisely, the “polytopes”; thereare quasiconvex maps (real valued maps whose sublevel sets are convex), quasiconcave maps (maps whose negativeis quasiconvex) and quasiaffine maps (simultaneously quasiconvex and quasiconcave) but, in this context, one couldask what a convex map should be; and, even in a framework like geodesic convexity, where a natural definition ofconvex maps is available, the existence of nonconstant convex maps might be problematic.

For an easily formulated, but apparently very difficult problem, on the convex hull of three points with respect tothe geodesic convexity on a Riemannian manifold see page 231 of [4].

Convexity—of maps—is an important and useful property in optimization and mathematical programming and onehas therefore been led to various generalizations of the concept of convex maps, as, for example, in [21,22] or [24].Convex sets play a very modest role, if they are defined at all, and when they are defined, as in [10], for example, theirproperties are not very well understood.

One could say that convexity has grown two branches, generalized convex sets and generalized convex maps, withpractically no interaction. B-convexity is one example of a generalized convexity which has both nice topological andfunctional properties.

Murota’s book [20], which is mainly on the functional side of generalized convexity, finds its motivation, and itsapplications, in discrete optimization. Coppel’s book [9] offers a beautiful axiomatic study of convexity.

The ideas and results presented in this paper owe a lot to the work of Maurice Van de Vel [26,27].Professor Valentin Gutev read a first draft of this paper. In that version Lemma 3.1 had a much more restricted

form and a rather long and difficult proof. He showed the author how to use Theorem 3.1 of [14]. Also, the proof ofTheorem 3.1 of this paper incorporates some of his suggestions.

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Continuous selections of multifunctions with weakly convex values

Grigorii E. Ivanov 1

Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny,Moscow region, Russia 141700

Received 20 July 2006; received in revised form 23 March 2007; accepted 23 March 2007

Abstract

We obtain some selection theorems for multifunctions with weakly convex values. For this purpose, some new properties ofweakly convex sets in a Hilbert space are investigated. We also present some examples showing the importance of various assump-tions in these selection theorems.© 2007 Elsevier B.V. All rights reserved.

MSC: 54C60; 54C65; 52A01

Keywords: Multifunctions; Continuous selection; Weakly convex set

One of the most known selection theorems was proved by Michael [1]. It claims that a lower semicontinuous(l.s.c.) multifunction on a paracompact space X with nonempty closed convex values in a Banach space Y admits acontinuous selection. Michael, Semenov, and Repovš [3–5] have shown that the assumption on convexity of the valuesmight be replaced by paraconvexity or functional paraconvexity. We propose another way to weaken the convexitycondition in selection theorems. Namely, we consider multifunctions with weakly convex values.

The convexity of a closed set A can be obviously reformulated as the property

P1: for any x0 ∈ A, x1 ∈ A there exists a point x ∈ A ∩ [x0;x1] that differs from the points x0, x1.

Following Vial [6], the definition of a weakly convex set with respect to (w.r.t.) R > 0 can be obtained from P1 byreplacing the straight line segment [x0;x1] by a strongly convex segment DR(x0;x1) (see Definitions 1 and 2).

In [6], some properties of weakly convex sets in an Euclidean finite dimensional space are studied. Similar andother properties of weakly convex sets in a Hilbert space have been studied in [7–9]. The class of weakly convex setsincludes both all convex sets and all solid-smooth sets (see Definition 5 and Proposition 1). If a set A in a Hilbertspace is closed and weakly convex w.r.t. R, then for any x from R-neighborhood of A there exists the unique metricprojection of x on A (see Proposition 2). The weakly convex sets, as well as strongly convex sets, admit interestingapplications in the optimal control theory and the differential games theory [9].

E-mail address: [email protected] Supported by the Russian Federation President grant MD-10079.2006.1.


852 G.E. Ivanov / Topology and its Applications 155 (2008) 851–857

In general, an l.s.c. multifunction with nonempty closed weakly convex values may have no continuous selection.Some sufficient conditions for continuous selections to exist are discussed below.

In Section 1, we give definitions and discuss the relation between weakly convex and paraconvex sets. In Section 2selection Theorems 3–6 are formulated. In each of the theorems we assume that the multifunction is l.s.c. and its valuesare nonempty closed and weakly convex. Besides, in Theorem 3 we assume that the values of the multifunction belongto balls with sufficiently small radii; in Theorem 4 we assume that the multifunction admits continuous ε-selectionwith sufficiently small ε; in Theorem 5 we assume that the multifunction acts from an (n + 1)-dimensional para-compact space and its values are n-connected. Finally, in Theorem 6 we assume that the multifunction is uniformlycontinuous and acts from a uniformly functional contractible set.

At the beginning of Section 3, we show that the class of weakly convex sets is rather wide; Propositions 1–3 givesome typical examples of weakly convex sets. Next, we present some examples of multifunctions to show that theassumptions of Theorems 3–6 are essential. At the end of Section 3, we discuss necessary conditions and sufficientconditions for a set to be uniformly functional contractible. Selection Theorems 4–6 are proved in Section 4.

1. Weakly convex and paraconvex sets

Let H be a Hilbert space. We denote by intA, clA, ∂A, respectively, the interior, the closure and the boundary ofthe set A in a topological space.

Definition 1. Let x0, x1 be points in a normed space E; suppose that R � ‖x1−x0‖2 . A strongly convex segment

DR(x0, x1) is the intersection of all balls BR(c) containing both x0 and x1:

DR(x0, x1) =⋂

c∈E: {x0,x1}⊂BR(c)

BR(c),

where

BR(c) = {x ∈ E: ‖x − c‖ � R

}.

Note that the strongly convex segment DR(x0, x1) tends to the segment [x0;x1] in the sense of the Hausdorff metricas R → +∞.

J.-Ph. Vial [6] have introduced the following definition.

Definition 2. A set A in a normed space is weakly convex w.r.t. R > 0 if for all points x0, x1 ∈ A such that 0 <

‖x1 − x0‖ < 2R, there exists a point x ∈ DR(x0, x1) ∩ A that differs from the points x0, x1.

Recall some elementary properties of weakly convex sets.

(1) If a set A in a normed space is weakly convex w.r.t. R, then it is weakly convex w.r.t. any r ∈ (0;R).(2) If a set A in a normed space is convex, then it is weakly convex w.r.t. any R > 0.(3) If a set A ⊂ H is closed and weakly convex w.r.t. any R > 0, then it is convex.

We denote by dist(y,A) the distance from the point y ∈ E to the set A ⊂ E:

dist(y,A) = infx∈A

‖x − y‖.

Definition 3. (See Michael [3].) Let A be a set in a normed space E; λ ∈ [0;1). Suppose that

dist(y,A) � λγ

for any γ > 0, x ∈ E and for any y from the convex hull of A ∩ intBγ (x). Then the set A is called a λ-paraconvexset.

Theorem 1. (See [9, Theorem 3.3.2].) Assume that a set A ⊂ H is closed and weakly convex w.r.t. R. Let r ∈ (0;R)

and c ∈H be such that A ⊂ Br (c). Then the set A is λ-paraconvex with λ = r/R.

G.E. Ivanov / Topology and its Applications 155 (2008) 851–857 853

Remark 1. A closed and weakly convex set is not necessary paraconvex. For example, the set A = {x ∈ Rn: ‖x‖ � 1}

is closed and weakly convex w.r.t. 1 but is not paraconvex.

Remark 2. A paraconvex set is not necessary weakly convex. For example, the set

A = {(x, y) ∈ R

2: x � 0 or y � 0}

is λ-paraconvex with λ = 1√2

but is not weakly convex w.r.t. any R > 0.

2. Selection theorems

We shall use the following theorem.

Theorem 2. (See Michael [3].) Let X be a paracompact space, Y be a Banach space, F :X → 2Y be an l.s.c.multifunction, λ ∈ [0;1). Suppose that for any x ∈ X the value F(x) is nonempty, closed, and λ-paraconvex. Then F

admits a continuous selection.

Combining Theorems 1 and 2, we obtain the following selection theorem.

Theorem 3. Let X be a paracompact space, F :X → 2H be an l.s.c. multifunction, 0 < r < R. Suppose that for anyx ∈ X the value F(x) is nonempty, closed, weakly convex w.r.t. R, and contained in some ball with radius r . Then F

admits a continuous selection.

As a standard corollary of Theorem 3 we obtain the following lemma.

Lemma 1. Suppose that a set A ⊂ H is closed, weakly convex w.r.t. R, and contained in some ball with radius r < R.Then A is contractible in A.

Theorem 4. Let X be a paracompact space, 0 < ε < R. Let F :X → 2H be an l.s.c. multifunction such that for allx ∈ X the value F(x) is nonempty, closed, and weakly convex w.r.t. R. Let f :X → H be an ε-selection of F , i.e.dist(f (x),F (x)) < ε for all x ∈ X. Then F admits a continuous selection.

Below we prove two more selection theorems. As usual, we assume that the values of the multifunction are weaklyconvex w.r.t. R. Unlike Theorem 3, the values may not be contained in balls with radius r < R.

Theorem 5. Let X be a paracompact space; dimX = n+1; R > 0. Let F :X → 2H be an l.s.c. multifunction. Supposethat F(x) is nonempty, n-connected, closed, and weakly convex w.r.t. R for all x ∈ X. Then F admits a continuousselection.

In the following selection theorem we does not require that the values of the multifunction are bounded (as inTheorem 3) or n-connected (as in Theorem 5). Instead of these, we require the uniform continuity of the multifunctionon a uniformly functional contractible set.

Denote R+ = [0;+∞).

Definition 4. A set X in a topological vector space E is uniformly functional contractible if there exists a point x0 ∈ X

and continuous functions T :X → R+, ϕ :X × R+ → X such that

T(ϕ(x, t)

)� t, ϕ

(x,T (x)

) = x, ϕ(x,0) = x0 ∀t ∈ R+ ∀x ∈ X, (1)

and the function t → ϕ(x, t) is uniformly continuous w.r.t. (x, t) ∈ X × R+, i.e.

for any neighborhood V ⊂ E of origin there exists δ > 0 such that for any x ∈ X andfor any t, t ′ ∈ R+ such that |t − t ′| � δ we have ϕ(x, t ′) − ϕ(x, t) ∈ V .

}(2)


Theorem 6. Let X be a uniformly functional contractible set in a paracompact vector space E. Let F :X → 2H bea uniformly continuous (in the sense of the Hausdorff metric) multifunction. Suppose that F(x) is nonempty, closed,and weakly convex w.r.t. R for all x ∈ X. Then F admits a continuous selection.

3. Examples

Propositions 1–3 provide some typical examples of weakly convex sets.

Definition 5. A set A ⊂ H is solid-smooth w.r.t. L if

(1) the set A is solid (i.e. cl intA = clA and int clA = intA) and(2) at every point x ∈ ∂A there exists the tangent hyperplane for A and, moreover, the vector field of unit normals

n(x) for A is Lipschitz continuous with constant L:∥∥n(x1) − n(x2)

∥∥ � L‖x1 − x2‖ ∀x1, x2 ∈ ∂A.

Proposition 1. (See [9, Theorem 1.10.2].) Suppose that a set A ⊂ H is closed, solid, and A �= H. The followingconditions are equivalent:

(1) the set A is solid-smooth w.r.t. L; and(2) the set A and its complement H \ A are weakly convex w.r.t. 1/L.

Thus, the class of weakly convex sets includes both all convex sets and all closed solid-smooth sets.

Definition 6. Let A be a set in a normed space E, x ∈ E. A point π ∈ A such that ‖x − π‖ = dist(x,A) is called ametric projection of x on A.

Proposition 2. (See [8, Theorem 1]; [9, Theorem 1.7.4].) Let a set A ⊂ H be closed and weakly convex w.r.t. R. ThenA satisfies the property

P2: metric projection to A is singlevalued over an R-neighborhood of A.

Moreover, the property P2 is sufficient for a locally compact set A ⊂ H to be weakly convex w.r.t. R.

The open question: Is P2 sufficient for any set in a Hilbert space to be weakly convex w.r.t. R?

Proposition 3. (See [9, Theorem 1.7.2].) Let a set A ⊂ H be closed, R > 0. The following conditions are equivalent:

(1) A is weakly convex w.r.t. R; and(2) if x /∈ A and π is a metric projection of the point x on A, then

intBR

(π + R

‖x − π‖ (x − π)

)∩ A = ∅.

The following examples show that the assumptions of Theorems 3, 5, 6 are important.In Theorem 3, we assumed that for every x ∈ X the value F(x) is contained in a ball with radius r < R. Exam-

ples 1–3 imply that this assumption is essential.The assumption of Theorem 5 requires n-connectedness of the values of F . This condition is essential. In particular,

if dimX = 1, then the linear connectedness (0-connectedness) is needed. This is illustrated by Examples 1 and 2. IfdimX = 2, then the 1-connectedness of the values is required, which follows from Example 3.

Example 1 shows that in Theorem 6 the uniform continuity of the multifunction F cannot be substituted by l.s.c.(as in Theorems 3–5). Example 2 illustrates the importance of the assumption in Theorem 6 that the set X is uniformlyfunctional contractible.


Example 1. Let the multifunction F : [0;3] → 2R be given by

F(x) ={ {0}, x ∈ [0;1],

{0;1}, x ∈ (1;2),

{1}, x ∈ [2;3].The multifunction F is l.s.c.; its values are nonempty closed and weakly convex w.r.t. 1/2. Nevertheless, the multi-function F does not admit a continuous selection.

Example 2. Let X = {z ∈ C: |z| = 1}, F(z) = √z. For every z ∈ X the set F(z) consists of two points in C. The

multifunction F :X → 2C is Lipschitz continuous with constant 1/2 in the sense of the Hausdorff metric. The valuesof F are nonempty closed and weakly convex w.r.t. 1. Nevertheless, the multifunction F does not admit a continuousselection.

Example 3. Let B1 be a unit ball in R2, and ∂B1 be its boundary. Let the multifunction F :B1 → 2R

2be given by

F(x) ={ {x}, ‖x‖ � 1,

∂B1, ‖x‖ < 1.

Then the multifunction F is l.s.c., and its values are nonempty closed, connected, and weakly convex. Nevertheless,the multifunction F does not admit a continuous selection because the boundary of the ball is not its retract.

Now we present some necessary conditions and sufficient conditions for a set to be uniformly functional con-tractible.

Lemma 2. Let X be a uniformly functional contractible set in a topological vector space E. Then X is contractiblein itself in the sense of topology in E.

Proof. By Definition 4, there exist a point x0 ∈ X and continuous functions T :X → R+, ϕ :X × R+ → X such that(1), (2) are valid. To each x ∈ X and t ∈ [0;1] we assign f (x, t) = ϕ(x, t ·T (x)). Then the function f :X×[0;1] → X

is continuous, and we have f (x,0) = x0, f (x,1) = x for all x ∈ X. �Lemma 3. Let X be a star set in a normed space E, i.e. there exists the center x0 ∈ X such that for any point x ∈ X

we have [x0;x] ⊂ X. Then X is uniformly functional contractible.

Proof. The functions

T (x) = ‖x − x0‖, ϕ(x, t) ={

x0 + tx − x0

‖x − x0‖ , t � ‖x − x0‖,x, t > ‖x − x0‖

satisfy Definition 4. �Lemma 4. Let EX,EY be topological vector spaces, X ⊂ EX , Y ⊂ EY . Let g :X → Y be a uniformly continuoushomeomorphism. Suppose that X is uniformly functional contractible. Then Y is uniformly functional contractible.

Proof. By Definition 4, there exist a point x0 ∈ X and continuous functions T :X → R+, ϕ :X × R+ → X such that(1), (2) are valid. We denote y0 = g(x0). To each y ∈ Y , t ∈ R+ we assign

T1(y) = T(g−1(y)

), ϕ1(y, t) = g

(ϕ(g−1(y), t

)).

Then the functions T1 :Y → R+ and ϕ1 :Y × R+ → Y satisfy conditions (1), (2) with x0, x,X,T ,ϕ being substitutedby y0, y,Y,T1, ϕ1, respectively. �


4. Proofs of Theorems 4–6

Combining Definitions 1 and 2 we obtain the following lemma.

Lemma 5. Let A be a set in a normed space E, x ∈ E. Suppose that A is weakly convex w.r.t. R, r ∈ (0;R], andA ∩ Br (x) �= ∅. Then A ∩ Br (x) is weakly convex w.r.t. R.

Lemma 6. Let a set A ⊂ H be closed and weakly convex w.r.t. R. Let a number ε and a point x ∈ H be such thatdist(x,A) < ε � R. Then

A ∩ Bε(x) = cl(A ∩ intBε(x)

).

Proof. Since the set A ∩ Bε(x) is closed, cl(A ∩ intBε(x)) ⊂ A ∩ Bε(x).Let us prove the backward inclusion. Let y ∈ A ∩ Bε(x). By Proposition 2, there exists the metric projection π of

the point x onto the set A. Inequalities ‖y − x‖ � ε, ‖π − x‖ < ε and Definition 1 imply that(Dε(π,y) \ {y}) ⊂ intBε(x). (3)

By [7, Theorem 4.1] set DR(π,y) ∩ A is connected. Consequently, there exists a sequence {yk} ⊂ DR(π,y) ∩ A

such that yk → y as k → ∞ and yk �= y for all k ∈ N. Therefore, it follows from inclusions (3) and DR(π,y) ⊂Dε(π,y) that yk ∈ A ∩ intBε(x) for all k ∈ N. Thus, y ∈ cl(A ∩ intBε(x)). �Proof of Theorem 4. To each x ∈ X we assign G(x) = F(x) ∩ Bε(f (x)). Since the multifunction F is l.s.c. andf is continuous, the multifunction x → F(x) ∩ intBε(f (x)) is l.s.c. Therefore, the multifunction x → cl(F (x) ∩intBε(f (x))) is l.s.c.

By Lemma 6, we have G(x) = cl(F (x)∩ intBε(f (x))) for all x ∈ X. Consequently, the multifunction G :X → 2H

is l.s.c.By Lemma 5, for any x ∈ X the set G(x) is weakly convex w.r.t. R. Therefore, by Theorem 3 there exists a

continuous selection of the multifunctions G and F as well. �Lemma 7. Let S be a family of closed weakly convex w.r.t. R sets A ⊂ H. Then for any n ∈ N ∪ {0}, the family S isequi-LCn.

Proof. Let x ∈ H, and U be a neighborhood of x in H. Consider an open ball V ⊂ H with the center x and radiusr � R such that clV ⊂ U . It follows from Lemma 5 that for any A ∈ S the set A ∩ clV is either empty or weaklyconvex w.r.t. R. In the former case, the set A∩ clV is contractible in itself by Lemma 1. Therefore, for any A ∈ S theset A ∩ V is either empty or contractible in A ∩ U . �Proof of Theorem 5. Apply Michael’s finite-dimensional theorem [2] and Lemma 7. �Proof of Theorem 6. Let us fix a number r ∈ (0;R). Since the multifunction F :X → 2H is uniformly continuous inthe sense of the Hausdorff metric, there exists the neighborhood V ⊂ E of origin such that

x1 − x2 ∈ V ⇒ h(F(x1),F (x2)

)� r ∀x1, x2 ∈ X,

where h(F (x1),F (x2)) is the Hausdorff distance between the sets F(x1) and F(x2).Since the set X is uniformly functional contractible, there exists a point x0 ∈ X and continuous functions T :X →

R+, ϕ :X × R+ → X such that the conditions (1), (2) of Definition 4 are valid. It follows from the condition (2) thatthere exists δ > 0 such that ϕ(x, t ′) − ϕ(x, t) ∈ V for all x ∈ X and all t, t ′ ∈ R+ such that |t − t ′| � δ. Hence,

h(F

(ϕ(x, t ′)

),F

(ϕ(x, t)

))� r ∀x ∈ X ∀t, t ′ ∈ R+: |t − t ′| � δ. (4)

To each k ∈ N ∪ {0} we assign the closed set

Xk = {x ∈ X: T (x) � kδ

}.


For every x ∈ X0 we have T (x) = 0; consequently, using (1), we obtain x = ϕ(x,T (x)) = ϕ(x,0) = x0. Thus,X0 = {x0}.

Let us fix a point y0 ∈ F(x0). We define f (x0) = y0. Thus, we defined the continuous selection f :X0 → H of themultifunction F :X0 → 2H.

Let us extend the continuous selection f step by step on sets X1 ⊂ X2 ⊂ · · · ⊂ Xk ⊂ · · · .Assume that for some k ∈ N ∪ {0} the continuous selection f :Xk → H of the multifunction F :Xk → 2H is

defined. Using condition (1), we get T (ϕ(x, kδ)) � kδ for all x ∈ X. Therefore, the value f (ϕ(x, kδ)) is well defined.To each element x ∈ Xk+1 we assign

gk(x) ={

f (x), T (x) � kδ,

f(ϕ(x, kδ)

), kδ < T (x) � (k + 1)δ,

Gk(x) ={ {f (x)}, T (x) � kδ,

F (x), kδ < T (x) � (k + 1)δ.

If T (x) = kδ, then the condition (1) implies that ϕ(x, kδ) = x. Therefore, using the continuity of functions f,ϕ,T ,we get the continuity of the function gk :Xk+1 → H.

Since the functions f,T are continuous and the multifunction F is l.s.c., the multifunction Gk :Xk+1 → 2H is l.s.c.on the set Xk+1.

Now by the equality ϕ(x,T (x)) = x and condition (4) we obtain

h(F

(ϕ(x, kδ)

),F (x)

)� r ∀x ∈ X: kδ < T (x) � (k + 1)δ.

Hence, using the inclusion

gk(x) ∈{

F(x), T (x) � kδ,

F(ϕ(x, kδ)

), kδ < T (x) � (k + 1)δ,

we have

dist(gk(x),F (x)

)� r ∀x ∈ Xk+1.

Now by Theorem 4 we obtain a continuous selection of the multifunction Gk :Xk+1 → 2H. Thus, the continuousselection f can be extended on the set Xk+1. By induction we get the selection f of the multifunction F on thewhole X = ⋃

k∈NXk . Since Xk ⊂ intXk+1 and f is continuous on Xk+1 for all k ∈ N, we have that f is continuous

on X. �Acknowledgements

I express my thanks to D. Repovš and P.V. Semenov for their very useful suggestions.

References

[1] E. Michael, Continuous selections. I, Ann. of Math. 63 (1956) 361–382.[2] E. Michael, Continuous selections. II, Ann. of Math. 64 (1956) 562–580.[3] E. Michael, Paraconvex sets, Scand. Math. 7 (1959) 372–376.[4] P.V. Semenov, Functionally paraconvex sets, Mat. Zametki 54 (6) (1993) 74–81 (in Russian); English transl.: Math. Notes 54 (6) (1993) 1236–

1240.[5] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, in: Math. and its Appl., vol. 455, Kluwer Acad. Publ., 1998,

pp. 1–369.[6] J.-Ph. Vial, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (2) (1982) 231–259.[7] G.E. Ivanov, Weak convexity in the senses of Vial and Efimov–Stechkin, Izvestiya AN, Ser. Mat. 69 (6) (2005) 35–60 (in Russian); English

transl.: Izv. Math. 69 (6) (2005) 1113–1135.[8] M.V. Balashov, G.E. Ivanov, Properties of metric projection on the weakly convex by Vial set and parametrization of set-valued mapping with

weakly convex images, Mat. Zametki 80 (4) (2006) 483–489 (in Russian); English transl.: Math. Notes 80 (4) (2006) 461–467.[9] G.E. Ivanov, Weakly Convex Sets and Functions: Theory and Applications, Fizmatlit, Moscow, 2006, 352 p. (in Russian).



A note on convex Gδ-subsets of Banach spaces

I. Namioka ∗, E. Michael

Department of Mathematics, University of Washington, P.O. Box 354350, Seattle, WA 98195-4350, USA

Received 20 December 2006; received in revised form 11 August 2007; accepted 11 August 2007

Abstract

We give selection characterizations of those convex Gδ-subsets Y of a Banach space E with the property that the closed (in Y )convex hull of each compact subset of Y is compact.© 2007 Elsevier B.V. All rights reserved.

MSC: primary 54C65, 46B50; secondary 46A55, 46B99

Keywords: Continuous selections; Closed convex hulls of compact sets

1. Introduction

The principal purpose of this note is to establish the following result, where F(Y ) denotes the collection of allnonempty closed subsets of Y .

Theorem 1.1. Let Y be a convex Gδ-subset of a Banach space E. Then the following conditions are equivalent:

(a) If K ⊂ Y is compact, then so is the closed (in Y ) convex hull of K .(b) If X is paracompact, and if ϕ :X → F(Y ) is lsc (= lower semicontinuous) with every ϕ(x) convex, then ϕ has a

continuous selection.(c) Same as (b), but with X assumed to be compact metrizable.

Remark 1.2. It is well known that the conditions in Theorem 1.1 are always satisfied if the convex set Y is closed inE (i.e. complete in the norm metric). The question whether (b) is satisfied even when the convex set Y is only a Gδ inE (i.e. completely metrizable) was raised in [5]. It was answered, in the negative, by Filippov in [2], where he showedthat an example of a convex Gδ-subset Y of a Banach space E in [10, p. 323] does not even satisfy (c), essentiallybecause it does not satisfy (a). Generalizing Filippov’s proof, we show in Section 4 that (c) always implies (a).

The implication (a) ⇒ (b) is proved in Section 2, (b) ⇒ (c) is obvious, and (c) ⇒ (a) is proved in Section 4 withthe aid of some preliminary material in Section 3. Some concluding remarks are in Section 5.

* Corresponding author.E-mail addresses: [email protected] (I. Namioka), [email protected] (E. Michael).


I. Namioka, E. Michael / Topology and its Applications 155 (2008) 858–860 859

2. Proof of (a) ⇒ (b) in Theorem 1.1

This result appeared, with an outlined proof, in [5, Remark 3.11, p. 277]. For completeness, we now give that proofwith a bit more detail.

Assume (a), and let ϕ :X → F(Y ) be as in (b). We must show that ϕ has a continuous selection.Since E is completely metrizable, so is its Gδ subset Y . Hence, by [4, Theorem 1.1] or [1, Theorem 3.2], there

exists a lscψ : X → F(Y ) with ψ(x) ⊂ ϕ(x) and with ψ(x) compact for all x ∈ X. For x ∈ X, let θ(x) be the closed(in Y ) convex hull of ψ(x). By (a), each θ(x) is compact, and thus also closed in E. Since ψ is lsc, so is θ : X → F(E)

by [3, Propositions 2.6 and 2.3], and hence θ has a continuous selection f by [3, Theorem 3.2′′]. But θ(x) ⊂ ϕ(x) foreach x ∈ X because ϕ(x) is convex and closed in Y , so f is also a selection for ϕ.

3. Two preliminary facts

Here we record two facts which will be needed in Section 4. Our first result, Fact 3.1, is equivalent to the BrouwerFixed Point Theorem.

Fact 3.1. (See, for example, [6].) Let Δ be a simplex, and let g : Δ → Δ be a continuous map such that g(Δ′) ⊂ Δ′for each face Δ′ of Δ. Then g(Δ) = Δ.

For our second result in Fact 3.2, we need some more terminology.Let K be a nonempty compact Hausdorff space. A probability measure on K is a continuous linear functional μ on

C(K), the Banach space of all continuous real-valued functions on K with the supremum norm, such that μ(1) = 1and such that μ(f ) � 0 for each f ∈ C(K) with f � 0. Let P(K) (or simply P ) be the subset of the dual C(K)∗ ofC(K) consisting of all probability measures on K with the topology induces from the weak∗ topology for C(X)∗, i.e.the weakest topology on C(K)∗ that makes the map ν �→ ν(f ) from C(K)∗ to R continuous for each f ∈ C(K). Thespace P is compact and convex, and it is metrizable if K is metrizable. For each y ∈ K , let δy denote the element ofP given by δy(f ) = f (y) for each f ∈ C(K). Then the map y �→ δy is a homeomorphic embedding of K into P .

By the Riesz Representation Theorem (e.g., Rudin [8, Theorem 2.14]), each μ ∈ P is uniquely represented bya regular Borel probability measure on K denoted again by μ. The relationship between these two aspects of μ isexpressed as follows: For each open subset U of K ,

μ(U) = sup{μ(f ): f ∈ C(K, I), σ (f ) ⊂ U

}, (1)

where I = [0,1] and σ(f ) = {y ∈ K: f (y) = 0}. Let U = {U ⊂ K: U open in K,μ(U) = 0}. Then, since μ is reg-ular, μ(

⋃U) = 0. Define the support S(μ) of μ by S(μ) = K\⋃

U . Then S(μ) is a nonempty closed subset of K ,and it is characterized by the following property: For each open subset U of K ,

μ(U) = 0 ⇔ U ∩ S(μ) = ∅. (2)

The following fact is used by Filippov [2], and is a special case of a very abstract theorem of E.V. Shchepin [9].For the convenience of the reader, we now give a simple proof.

Fact 3.2. Using the notation above, the support map S : P →F(K), given by μ �→ S(μ), is lsc.

Proof. Let U be open in K . We must show that {μ ∈ P : S(μ) ∩ U = ∅} is open in P . By (2) and (1),{μ ∈ P : S(μ) ∩ U = ∅} = {

μ ∈ P : μ(U) > 0} = {

μ ∈ P : μ(f ) > 0 for some f ∈ C(K, I) with σ(f ) ⊂ U},

and the last set is open in P because μ �→ μ(f ) is continuous for each f ∈ C(K, I). �4. Proof of (c) ⇒ (a) in Theorem 1.1

Assume (c), and let K ⊂ Y be as in (a). We must show that the closure of convK in Y is compact.

860 I. Namioka, E. Michael / Topology and its Applications 155 (2008) 858–860

As in Section 3, let P ⊂ C(K)∗ be the space of all probability measures on K with the weak∗ topology. ThenP is compact, metrizable and convex. Let S : P → F(K) be the support function, and, as in Filippov [2], defineϕ : P →F(Y ) by

ϕ(μ) = (convS(μ)

)(closure in Y).

Since the map S is lsc by Fact 3.2, so is the map ϕ by [3, Propositions 2.6 and 2.3]. Since Y is convex, so is everyϕ(μ). Hence ϕ has a continuous selection s :P → Y by (c). We will show that convK ⊂ s(P ). That will finish theproof, for, since s(P ) is compact and hence closed in Y , the closure of convK in Y is contained in s(P ) and thereforecompact.

In the following, we use convY and convP for the convex hulls in Y and P , respectively. So let y0 ∈ convY K and letus show that y0 ∈ s(P ). Pick a finite A ⊂ K such that y0 ∈ convY A, and we may assume that A is minimal with respectto this property. Then A is affinely independent (see e.g. Rudin [7, Lemma on p. 73]). Now convP {δy : y ∈ A} is asimplex ΔA ⊂ P with vertices {δy : y ∈ A}, and the map A → ΔA given by y �→ δy extends to the affine-isomorphismh : convY A → ΔA. Note that a face of ΔA is a subset ΔB of ΔA of the form ΔB = convP h(B) = convP {δy : y ∈B}, where ∅ = B ⊂ A. Suppose μ ∈ ΔB with ∅ = B ⊂ A. Then S(μ) ⊂ B and therefore s(μ) ∈ ϕ(μ) ⊂ convY B .Consequently s(ΔB) ⊂ convY B . Then

h ◦ s(ΔB) ⊂ h(convY B) = convP

(h(B)

) = ΔB,

so the map h ◦ s|ΔA:ΔA → ΔA satisfies the hypothesis of Fact 3.1. It follows that h(s(ΔA)) = ΔA = h(convY A), so

s(ΔA) = convY A because h is one-to-one. In particular, y0 ∈ convY A = s(ΔA) ⊂ s(P ). �5. Concluding remarks

Remark 5.1. The implication (c) ⇒ (a) in Theorem 1.1 remains valid, with unchanged proof, even without assumingthat Y is a Gδ in E.

Remark 5.2. Theorem 1.1 remains valid, with essentially the same proof, without assuming that Y ⊂ E is convex,provided (a) is modified by also requiring that K ⊂ C for some closed (in Y ) convex C ⊂ Y . (If Y is convex, one cansimply take C = Y .) Thus, in particular, the equivalence (b) ⇔ (c) in Theorem 1.1 remains valid even when Y is notassumed to be convex.

Our final remark deals with analogs of conditions (a) and (c) in Theorem 1.1 when E = Rn.

Remark 5.3. Suppose E = Rn. Then:

(a′) If K ⊂ E is compact, so is convK . (See [7, Theorem 3.25].)(c′) If X is perfectly normal, and if ϕ is a l.s.c. map from X to the collection of (not necessarily closed) nonempty

convex subsets of E, then ϕ has a continuous selection. (See [3, Theorem 3.1′′′ (c) and lines 10–12 on p. 367].1)

References

[1] M.M. Choban, E. Michael, Representing spaces as images of zero-dimensional spaces, Topology Appl. 49 (1993) 217–220.[2] V.V. Filippov, On a question of E.A. Michael, Comment. Math. Univ. Carolinae 45 (2004) 735–737.[3] E. Michael, Continuous selection I, Ann. of Math. 63 (1956) 361–382.[4] E. Michael, A theorem on semi-continuous set-valued functions, Duke Math. J. 26 (1959) 647–652.[5] E. Michael, Some problems, in: Open Problems in Topology, North-Holland, Amsterdam, 1990, pp. 271–278.[6] I. Namioka, On certain onto maps, Can. J. Math. 14 (1962) 461–466.[7] W. Rudin, Functional Analysis, McGraw–Hill Book Company, New York, 1973.[8] W. Rudin, Real and Complex Analysis, third ed., McGraw–Hill Book Company, New York, 1987.[9] E.V. Shchepin, Functors and uncountable degrees of compacta (Russian), Uspekhi Mat. Nauk. 36 (1981) 3–62, 255.

[10] H.v. Weizsäcker, A note on infinite-dimensional convex sets, Math. Scand. 38 (1976) 321–324.

1 There is a misprint in [3, Theorem 3.1′′′ (c)], where K(Y ) should read D(Y ).



Applications of Michael’s selection theoremsto fixed point theory

Sehie Park

The National Academy of Sciences and Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea

Received 24 May 2006; received in revised form 9 February 2007; accepted 9 February 2007

Abstract

Applying some of Ernest Michael’s selection theorems, from recent fixed point theorems on u.s.c. multimaps, we deduce gener-alizations of the classical Bolzano theorem, several fixed point theorems on multimaps defined on almost convex sets, almost fixedpoint theorems, coincidence theorems, and collectively fixed point theorems. These results are related mainly to Michael maps,that is, l.s.c. multimaps having nonempty closed convex values.© 2007 Elsevier B.V. All rights reserved.

Keywords: t.v.s.; Multimap (map); Selection; Almost convex set; Almost fixed point property; Multimap class B

1. Introduction

The fixed point theory of multimaps in topological vector spaces has numerous applications in many fields inmathematical sciences. This theory began with the celebrated Kakutani fixed point theorem in 1941 and, until recently,was mainly concerned with the class of upper semicontinuous multimaps with closed convex values (usually calledKakutani maps).

Recall that Ernest Michael’s groundbreaking theory of continuous selections on multimaps began in 1956 and wasmainly concerned with the class of lower semicontinuous multimaps with closed convex values (which will be calledMichael maps). However, there have appeared only relatively fewer fixed point theorems on this class of multimaps.For the details, we can consult with the monograph [23].

In our previous works [14,17], we unified fixed point theorems for the class of convex-valued upper semicontinuous(or more general) multimaps defined mainly on convex subsets of topological vector spaces. Recently, there haveappeared a number of fixed point theorems for new classes of multimaps defined on convex subsets of topologicalvector spaces. Moreover, in [20], we obtained a new fixed point theorem for the ‘better’ admissible class Bp definedon almost convex subsets of topological vector spaces.

In the present paper, we obtain new results mainly on Michael maps by combining some of Michael’s selectiontheorems with a number of recent results in the fixed point theory of multimaps in topological vector spaces. Conse-quently, we deduce generalizations of the classical Bolzano theorem, several fixed point theorems on Michael maps

E-mail addresses: [email protected], [email protected].


862 S. Park / Topology and its Applications 155 (2008) 861–870

defined on almost convex sets, almost fixed point theorems, coincidence theorems, and collectively fixed point theo-rems. These results are related mainly to Michael maps.

In Section 2, we introduce three principal selection theorems of Michael in [10–12], which we will need in thispaper, and obtain a simple generalization of Bolzano’s theorem by applying one of Michael’s theorems. Section 3 dealswith a unified fixed point theorem on convex-valued upper hemicontinuous multimaps and some of its consequences.We obtain another multi-valued version of Bolzano’s theorem and a fixed point theorem on Michael maps.

In Sections 4 and 5, we deduce fixed point theorems on Michael maps defined on almost convex sets and somealmost fixed point theorems. Section 6 deals with existence theorems of coincidence points of multimaps in the classB of multimaps with continuous functions or Michael maps. Finally in Section 7, we obtain some collectively fixedpoint theorems for families of Michael maps.

2. When Bolzano meets Michael

A multimap F : X � Y is a function from a set X into the set 2Y of nonempty subsets of Y ; that is, a function withthe values F(x) ⊂ Y for x ∈ X and the fibers F−(y) := {x ∈ X | y ∈ F(x)} for y ∈ Y . We use the term map instead ofmultimap. For A ⊂ X, let F(A) := ⋃{F(x) |x ∈ A}. For any B ⊂ Y , the (lower) inverse of B under F is defined by

F−(B) := {x ∈ X | F(x) ∩ B �= ∅}

.

For topological spaces X and Y , a map F : X � Y is said to be closed if its graph

Gr(F ) := {(x, y) | y ∈ F(x), x ∈ X

}

is closed in X × Y , and compact if F(X) is contained in a compact subset of Y .A map F : X � Y is said to be upper semicontinuous (u.s.c.) if, for each closed set B ⊂ Y , F−(B) is closed in X;

lower semicontinuous (l.s.c.) if, for each open set B ⊂ Y , F−(B) is open in X; and continuous if it is u.s.c. and l.s.c.If F : X � Y is u.s.c. with closed values and if Y is regular, then F is closed. The converse is true whenever Y is

compact.Usually, a Kakutani map is an u.s.c. map with closed convex values. In parallel, a l.s.c. map with closed convex

values will be called a Michael map in this paper.The following are principal theorems of [10–12]:

Michael’s Theorem 3.2′′. (1956, [10]) The following properties of a T1-space X are equivalent:

(a) X is paracompact.(b) If Y is a Banach space, then every lower semi-continuous carrier (map) for X to the family of nonempty, closed,

convex subsets of Y admits a continuous selection.

Michael’s Theorem 1.2. (1966, [11]) Let X be paracompact, and M a metrizable subset of a complete locally convexspace E. Let φ : X → 2M be l.s.c. and such that, for some metric on M, every φ(x) is complete. Then there exists acontinuous f : X → E such that f (x) ∈ ΓEφ(x) for every x ∈ X.

Here, ΓEA denotes the closed convex hull coEA of A in E. Recall that the completeness can be replaced by thecompactness of ΓEK for every compact K ⊂ M .

If S is a topological space and A ⊂ S, dimS A � 0 means that the covering dimension of T is � 0 for every setT ⊂ A which is closed in S; see Hurewicz and Wallman [6].

Michael’s Theorem 7.1. (1981, [12]) Let X be a paracompact space, Y a Fréchet space, Z ⊂ X with dimX Z � 0,C ⊂ X countable, and ϕ : X → 2Y l.s.c. such that ϕ(x) is closed in Y for x /∈ C and ϕ(x) is convex when x /∈ Z. Thenϕ has a continuous selection f : X → Y .

In this theorem, we adopted the version of Ben-El-Mechaiekh and Oudadess [1, Corollary 6].In closing this section, in order to illustrate the usefulness of Michael’s works, we state a generalization of the

following:

S. Park / Topology and its Applications 155 (2008) 861–870 863

Bolzano’s Theorem. (1817, [13]) Let f : [−r, r] → R be a continuous function satisfying the following boundarycondition:

x · f (x) > 0 for |x| = r.

Then there exists at least one solution x0 ∈ [−r, r] of the equation f (x) = 0.

Note that actually x0 ∈]−r, r[.Combining Michael’s Theorem 1.2 [11] and Bolzano’s Theorem, we immediately obtain

Theorem 2.1. Let α > 0 and F : [−α,α] � R be a Michael map satisfying the following boundary condition:

x · y > 0 for |x| = α and y ∈ F(x).

Then there exists at least one solution x0 ∈]−α,α[ of the inclusion 0 ∈ F(x).

Similarly, some other results in [13] or other works can be stated for Michael maps.

3. Descendants of Bolzano

A t.v.s. means a Hausdorff topological vector space E. Let E∗ denote the topological dual of E. Kakutani’s convex-valued u.s.c. multimaps are further extended as follows: For a subset X of a t.v.s. E, a map F : X � E is said to be

(i) upper demicontinuous (u.d.c.) if for each x ∈ X and open half-space H in E containing F(x), there exists anopen neighborhood N of x in X such that F(N) ⊂ H ;

(ii) upper hemicontinuous (u.h.c.) if for each h ∈ E∗ and for any real α, the set {x ∈ X | sup RehF(x) < α} is openin X; and

(iii) generalized u.h.c. if for each p ∈ {Reh | h ∈ E∗}, the set {x ∈ X | suppF(x) � p(x)} is compactly closed in X.

For the literature, see [16].In our earlier works [14,17], we unified a large number of generalizations of the Kakutani theorem to maps of the

above-mentioned types. In this section, combining Michael’s theorems and the main fixed point theorem in [14,17],we deduce another generalizations of the Bolzano theorem and new fixed point theorems.

According to Lassonde [9], a convex space X is a nonempty convex set with any topology that induces the Euclid-ean topology on the convex hulls of its finite subsets. A nonempty subset L of a convex space X is called a c-compactset if for each finite set N ⊂ X there is a compact convex set LN ⊂ X such that L ∪ N ⊂ LN .

Let cc(E) denote the set of nonempty closed convex subsets of a t.v.s. E and kc(E) the set of nonempty compactconvex subsets of E.

Let X be a nonempty convex subset of a vector space E. The algebraic boundary δE(X) of X in E is the set ofall x ∈ X for which there exists y ∈ E such that x + ry /∈ X for all r > 0. If E is a t.v.s., the topological boundaryBdX = BdE X of X is the complement of IntEX in the closure X. It is known that δE(X) ⊂ BdX and in generalδE(X) �= BdX.

Let X ⊂ E and x ∈ E. The inward and outward sets of X at x, IX(x) and OX(x), resp., are defined as follows:

IX(x) := x +⋃

r>0

r(X − x), OX(x) := x +⋃

r<0

r(X − x).

For p ∈ {Reh | h ∈ E∗} and U,V ⊂ E, let

dp(U,V ) = inf{∣∣p(u − v)

∣∣: u ∈ U,v ∈ V}.

The following is the main theorem in [14,17]:

Theorem 3.1. Let X be a convex space, L a c-compact subset of X,K a nonempty compact subset of X, E a t.v.s.containing X as a subset, and F a map satisfying either

(A) E∗ separates points of E and F : X → kc(E), or(B) E is locally convex and F : X → cc(E).


(I) Suppose that for each p ∈ {Re h | h ∈ E∗},(0) p|X is continuous on X;(1) Xp := {x ∈ X | infpF(x) � p(x)} is compactly closed in X;(2) dp(F (x), IX(x)) = 0 for every x ∈ K ∩ δE(X); and(3) dp(F (x), IL(x)) = 0 for every x ∈ X � K .Then there exists an x ∈ X such that x ∈ F(x).

(II) Suppose that for each p ∈ {Re h | h ∈ E∗},(0)′ p|X is continuous on X;(1)′ Xp := {x ∈ X | suppF(x) � p(x)} is compactly closed in X;(2)′ dp(F (x),OX(x)) = 0 for every x ∈ K ∩ δE(X); and(3)′ dp(F (x),OL(x)) = 0 for every x ∈ X � K .Then there exists an x ∈ X such that x ∈ F(x). Further, if F is u.h.c., then F(X) ⊃ X.

Remarks. 1. In Theorem 3.1, we do not require any concrete connection between topologies of X and E except (0).This is why there have appeared fixed point theorems on maps whose domains and ranges have different topologies.

2. If F is u.h.c. on each nonempty compact subset C of X, then F satisfies the “continuity” condition (1) for allp ∈ {Re h | h ∈ E∗}, but not conversely. Any map F satisfying (1) is generalized u.h.c.

3. The “boundary” condition (2) is equivalent to the following:

(2)′′ x ∈ K and p(x) = maxp(IX(x)) implies x ∈ Xp .

In fact, p(x) = maxp(X) is equivalent to p(x) = maxp(IX(x)).Moreover, the “boundary” condition (2)′′ is equivalent to the following:

(2)′′′ x ∈ K ∩ δE(X) and p(x) = maxp(IX(x)) implies x ∈ Xp .

4. The “coercivity” or “compactness” condition (3) is equivalent to the following:

(3)′′ x ∈ X � K and p(x) = maxp(IL(x)) implies x ∈ Xp .

5. For conditions (0)′–(3)′, facts similar to 1–4 hold. The property F(X) ⊃ X is called the surjectivity of F .

Recall that Theorem 3.1 unifies, improves and generalizes historically well-known fixed point theorems publishedin nearly 50 papers; see the diagrams and references in [14,16,17].

For a Kakutani map, Theorem 3.1 reduces to the following:

Corollary 3.2. Let X be a convex subset of a t.v.s. E, L a c-compact subset of X,K a nonempty compact subset of X,and F an u.s.c. map satisfying either

(A) E∗ separates points of E and F : X → kc(E), or(B) E is locally convex and F : X → cc(E).

(I) Suppose that(i) F(x) ∩ IX(x) �= ∅ for every x ∈ K ∩ δE(X); and

(ii) F(x) ∩ IL(x) �= ∅ for every x ∈ X � K .Then there exists an x ∈ X such that x ∈ F(x).

(II) Suppose that(i)′ F(x) ∩ OX(x) �= ∅ for every x ∈ K ∩ δE(X); and

(ii)′ F(x) ∩ OL(x) �= ∅ for every x ∈ X � K .Then there exists an x ∈ X such that x ∈ F(x) and F(X) ⊃ X.

For a single-valued map defined on a convex subset of a locally convex t.v.s., Theorem 3.1 reduces to the following:


Corollary 3.3. Let E be a locally convex t.v.s., X a convex subset of E, L a c-compact subset of X,K a nonemptycompact subset of X, and f : X → E a continuous function.

(I) Suppose that(i) f (x) ∈ IX(x) for every x ∈ K ∩ δE(X); and

(ii) f (x) ∈ IL(x) for every x ∈ X � K .Then there exists an x ∈ X such that x = f (x).

(II) Suppose that(i)′ f (x) ∈ OX(x) for every x ∈ K ∩ δE(X); and

(ii)′ f (x) ∈ OL(x) for every x ∈ X � K .Then there exists an x ∈ X such that x = f (x) and f (X) ⊃ X.

The following is another multi-valued version of Bolzano’s theorem:

Theorem 3.4. Let G : [−α,α] � R be a Kakutani map satisfying the following boundary condition:

if |x| = α, then x · y > 0 for some y ∈ G(x).

Then there exists at least one solution x0 ∈ [−α,α] of the inclusion 0 ∈ G(x).

Proof. Let F(x) := G(x) + x for each x ∈ [−α,α]. It suffices to show that F has a fixed point. We use Corolla-ry 3.2(II) with E = R and X = L = K = [−α,α]. Note that condition (ii)′ holds trivially. For x = α, α · y > 0 impliesy > 0 for some y ∈ G(α), and hence

F(α) ∩ OX(α) = (G(α) + α

)∩]α,∞[ �=∅.

Similarly, we have y < 0 for some y ∈ G(−α), and hence

F(−α) ∩ OX(−α) = (G(−α) − α

)∩]−∞,−α[ �=∅.

Hence condition (i)′ holds. Therefore the conclusion follows from Corollary 3.2(II). �Remark. Bolzano’s theorem follows from Theorem 3.4, and hence from Theorem 3.1. Now, instead of the Brouwerfixed point theorem in 1912, Bolzano’s theorem in 1817 can be regarded as the foremost ancestor of Theorem 3.1. Wewill meet another descendants of Bolzano’s.

In view of Michael’s Theorem 1.2 [11], we immediately deduce the following from Corollary 3.3:

Theorem 3.5. Let E be a completely metrizable locally convex t.v.s., X a convex subset of E, L a compact convexsubset of X,K a nonempty compact subset of X, and F : X � E a Michael map.

(I) Suppose that(i) F(x) ⊂ IX(x) for every x ∈ K ∩ δE(X); and

(ii) F(x) ⊂ IL(x) for every x ∈ X � K .Then there exists an x ∈ X such that x ∈ F(x).

(II) Suppose that(i)′ F(x) ⊂ OX(x) for every x ∈ K ∩ δE(X); and

(ii)′ F(x) ⊂ OL(x) for every x ∈ X � K .Then there exists an x ∈ X such that x ∈ F(x) and F(X) ⊃ X.

Recall that particular forms of Theorem 3.5 are due to Reich [22, Theorem 1.9] and Dugundji–Granas [2, Theo-rem 5.11.6]; see also [23].


4. Michael maps on almost convex sets

Let E be a t.v.s. with a base V of neighborhoods of 0. According to Himmelberg [4], a subset X of a t.v.s. E issaid to be almost convex if for any V ∈ V and for any finite subset A := {x1, x2, . . . , xn} of X, there exists a subsetB := {y1, y2, . . . , yn} of X such that yi − xi ∈ V for each i = 1,2, . . . , n and coB ⊂ X.

Here we give a well-known result due to Idzik [7]; see also [21]:

Theorem 4.1. Let X be an almost convex subset of a locally convex t.v.s. E and F : X � X a compact Kakutani map.Then F has a fixed point.

Remark. Theorem 4.1 for a convex X is due to Himmelberg [4] which contains results due to Brouwer, Schauder,Tychonoff, Hukuhara, Kakutani, Bohnenblust and Karlin, Fan, Glicksberg, and others; see [16].

Combining Michael’s Theorem 1.2 [11] with Theorem 4.1 for a single-valued function F and a convex subset X,we obtain the following:

Theorem 4.2. Let X be a convex subset of a locally convex t.v.s. E, and Y a compact metrizable subset of X. Then aMichael map S : X � Y has a fixed point.

Proof. Since Y is compact, by the well-known argument of Fournier and Granas [3], co Y is a σ -compact subsetof X and hence co Y is Lindelöf. Since co Y is regular as a subset of a t.v.s., we know that co Y is paracompact.Then, by Michael’s Theorem 1.2 [11], there exists a continuous function f : coY → E, where E is the completionof E, such that f (x) ∈ S(x) ⊂ Y ⊂ coY for all x ∈ coY . Note that f : coY → coY and f is compact. Therefore, byTheorem 4.1, f has a fixed point x0 ∈ coY ⊂ X; that is, x0 = f (x0) ∈ S(x0). This completes our proof. �Remarks. 1. Theorem 4.2 is due to Wu [24, Corollary 3] with slightly different proof. When X itself is compact andmetrizable, Theorem 4.2 reduces to Himmelberg et al. [5, Theorem 3].

2. Some particular forms of Theorem 4.2 are due to Kim and Lee, Zheng, and Zhang; see [15].

Similarly, we immediately deduce the following new result:

Theorem 4.3. Let E be a locally convex t.v.s. and X an almost convex metrizable subset of E. Then any compactMichael map F : X � X has a fixed point.

Proof. Since F(X) is a compact subset of X, we may regard it a compact metrizable subset of the completion E of E.Since X is a paracompact subset of E, by Michael’s Theorem 1.2 [11], F has a continuous selection f : X � F(X) ⊂X. Since f is compact, by Theorem 4.1, f has a fixed point x0 = f (x0) ∈ F(x0). This completes our proof. �Remark. When X is compact and convex, Theorem 4.3 reduces to Himmelberg et al. [5, Theorem 3].

Corollary 4.4. Let E be a normed vector space and X an almost convex subset of E. Then any compact Michael mapF : X � X has a fixed point.

Combining Michael’s Theorem 7.1 [12] with Theorem 4.1, we have the following:

Theorem 4.5. Let X be an almost convex subset of a Fréchet space E, Y a compact subset of X,Z ⊂ X with dimX Z �0, and C ⊂ X countable. Let T : X � Y be a l.s.c. map such that T (x) is closed for x /∈ C and T (x) is convex forx /∈ Z. Then T has a fixed point.

For normed vector spaces, we have the following form of Michael’s Theorem 7.1 [12]:


Lemma 4.6. Let X be a paracompact space, Z ⊂ X with dimX Z � 0, C ⊂ X countable, Y a normed vector space,and T : X � Y a l.s.c. map such that T (x) is complete for x /∈ C and T (x) is convex for x /∈ Z. Then T has acontinuous selection.

Proof. Without loss of generality, we may assume that Y is complete (for the conditions on T remain unchanged inthe completion of Y ). Now by applying Michael’s Theorem 7.1 [12], we have the conclusion. �

From Lemma 4.6 and Theorem 4.1, we have the following:

Theorem 4.7. Let X be an almost convex subset of a normed vector space, Z ⊂ X with dimX Z � 0, C ⊂ X countable,and T : X � X a compact l.s.c. map such that T (x) is closed for x /∈ C and T (x) is convex for x /∈ Z. Then T has afixed point.

Proof. Note that T (x) is compact and hence complete for each x /∈ C. Applying Lemma 4.6, T has a continuousselection f : X → X. Since f (X) ⊂ T (X) and T is compact, so is f . Therefore, by Theorem 4.1, f has a fixed pointx0 = f (x0) ∈ T (x0) ⊂ X. This completes our proof. �

For Z = C = ∅, Theorem 4.7 reduces to Corollary 4.4.Some results related to this section can be found in [18].

5. Almost fixed points

The celebrated KKM principle due to Knaster, Kuratowski, and Mazurkiewicz [8] in 1929 is as follows:

KKM principle. Let D be the set of vertices of a simplex S and F : D � S a map with closed [resp., open ] valuessuch that

coN ⊂ F(N) for each N ⊂ D.

Then⋂

z∈D F(z) �= ∅.

The KKM principle follows from the Sperner combinatorial lemma appeared in 1928 and was used to obtain one ofthe most direct proofs of the Brouwer fixed point theorem. Later, it was known that those three theorems are mutuallyequivalent. In fact, those three theorems are regarded as a sort of mathematical trinity. All are extremely importantand have many applications. Moreover, many important results in nonlinear functional analysis and other fields areknown to be equivalent to those three theorems. For details, see [16].

From the KKM principle, we obtained recently an almost fixed point theorem [19, Theorem 5.1], [20, Theorem 3.1]for u.s.c. or l.s.c. maps. The following is a particular form for close relatives of Kakutani maps or Michael maps:

Theorem 5.1. Let X be a convex subset of a locally convex t.v.s. E and T : X � X an u.s.c. [resp., a l.s.c.] map withconvex values such that T (X) is totally bounded. Then T has the almost fixed point property; that is, for each V ∈ V ,there exists an xV ∈ X such that T (xV ) ∩ (xV + V ) �= ∅.

Now we apply Theorem 5.1 to compact maps:

Theorem 5.2. Let X be a convex subset of a locally convex t.v.s. E and T : X � X a compact u.s.c. [resp., l.s.c.] mapwith convex values. Then there exists a point x0 ∈ X such that (x0, x0) ∈ Gr(T ).

Proof. For each element V ∈ V , there exist xV , yV ∈ X such that yV ∈ T (xV ) and yV ∈ xV + V . Since T (X) isrelatively compact in X, we may assume that the net yV converges to some x0 ∈ T (X). Then xV also converges to x0.Since (xV , yV ) ∈ Gr(T ), we have the conclusion. �

From Theorem 5.2, we have the following due to Himmelberg [4]:


Corollary 5.3. Let X be a convex subset of a locally convex t.v.s. E and T : X � X a compact Kakutani map. ThenT has a fixed point x0 ∈ T (x0).

Proof. Since T is u.s.c. with closed values and X is regular, the graph of T is closed in X × T (X). By Theorem 5.2,there exists a point x0 ∈ X such that (x0, x0) ∈ Gr(T ) = Gr(T ), and hence we have x0 ∈ T (x0). This completes ourproof. �6. Coincidence theorems

For maps F : X � Y and G : Y � X, a coincidence point (x0, y0) ∈ X × Y is the one satisfying y0 ∈ F(x0) andx0 ∈ G(y0) [that is, x0 ∈ X is a fixed point of G ◦ F : X � X].

An equivalent definition is as follows: For maps F : X � Y and G : X � Y , a coincidence point x0 ∈ X is the onesatisfying F(x0) ∩ G(x0) �= ∅ [that is, x0 ∈ X is a fixed point of G− ◦ F : X � X].

In this section, we derive existence theorems of coincidence points of maps in the class B with continuous functionsor Michael maps.

A polytope P in a subset X of a t.v.s. E is a homeomorphic image of a simplex.In 1996, the author introduced the ‘better’ admissible class B of maps defined on a subset X of a t.v.s. E into a

topological space Y as follows:

F ∈ B(X,Y ) ⇔ F : X � Y is a map such that, for each polytope P in X and for any continuousfunction f : F(P ) → P , the composition f (F |P ) : P � P has a fixed point.

Subclasses of B are classes of continuous functions C, the Kakutani maps K, the Aronszajn maps M (u.s.c. withRδ values), the acyclic map V (u.s.c. with compact acyclic values), the Powers maps Vc (finite compositions ofacyclic maps), the O’Neill maps N (continuous with values of one or m acyclic components, where m is fixed), theFan–Browder maps (codomains are convex sets), locally selectionable maps having convex values (codomains areconvex sets), the Simons maps Kc, the approachable maps A (whose domains are uniform spaces), admissible mapsof Górniewicz, σ -selectionable maps of Haddad and Lasry, permissible maps of Dzedzej, the class K

σc of Lassonde,

the class Vσc of Park et al., approximable maps of Ben-El-Mechaiekh and Idzik, and others. Those subclasses are

examples of the admissible class Aκc due to the author. Moreover, compact closed maps in the KKM class due to

Chang and Yen and in the s-KKM class due to Chang, Huang, and Jeng also belong to the class B. For references,see [20].

The following is a particular case of [20, Theorem 3.7]:

Theorem 6.1. Let X be an almost convex subset of a locally convex t.v.s. E and F ∈ B(X,X) a compact closed map.Then F has a fixed point.

From Theorem 6.1, we have the following:

Theorem 6.2. Let X be an almost convex subset of a locally convex t.v.s. E and Y a compact space. Let G ∈ B(X,Y )

be a closed map and f ∈ C(Y,X). Then G and f have a coincidence point.

Proof. We show that f ◦ G ∈ B(X,X). In fact, for any polytope P of X and any continuous g : (f ◦ G)(P ) → P ,the composition g((f ◦ G)|P ) = (g ◦ f )(G|P ) has a fixed point because G ∈ B(X,Y ). Moreover, since f is compactand continuous and G is closed, f ◦ G is closed and compact. Now, by Theorem 6.1, f ◦ G has a fixed point. �

In view of Michael’s Theorem 1.2 [11], we deduce the following from Theorem 6.2:

Theorem 6.3. Let X be an almost convex metrizable subset of a complete locally convex t.v.s. E and Y a compactspace. Let F : Y � X be a Michael map and G ∈ B(X,Y ) a closed map. Then F and G have a coincidence point.

In view of Michael’s Theorem 3.2′′ [10], we immediately have the following:


Corollary 6.4. Let X be an almost convex subset of a Banach space E and Y a compact space. Let F : Y � X be aMichael map and G ∈ B(X,Y ) a closed map. Then F and G have a coincidence point.

When Michael meets Kakutani, we have the following:

Corollary 6.5. Let X be an almost convex metrizable subset of a complete locally convex t.v.s. E1 and Y a compactsubset of a t.v.s. E2. Let F : Y � X be a Michael map and G : X � Y a Kakutani map. Then F and G have acoincidence point.

7. Collectively fixed points

In this section, we deduce some collectively fixed point theorems for families of Michael maps.Let {Xi}i∈I be a family of nonempty sets, and let i ∈ I be fixed. Let

X :=∏

j∈I

Xj and Xi :=∏

j∈I�{i}Xj .

If xi ∈ Xi and j ∈ I � {i}, let xij denote the j th coordinate of xi . If xi ∈ Xi and xi ∈ Xi , let [xi, xi] ∈ X be defined

as follows: its ith coordinate is xi and, for j �= i, the j th coordinate is xij . Therefore, any x ∈ X can be expressed as

x = [xi, xi] for any i ∈ I , where xi denotes the projection of x onto Xi .For A ⊂ X, xi ∈ Xi , and xi ∈ Xi , let

A(xi) := {yi ∈ Xi | [xi, yi] ∈ A

}and A(xi) := {

yi ∈ Xi | [yi, xi] ∈ A}.

The following are variants of known ones:

Theorem 7.1. Let {Xi}i∈I be a family of almost convex sets, each in a locally convex t.v.s. Ei , and Ti : X � Xi acompact Michael map for each i ∈ I . If X is metrizable, then there exists an x ∈ X such that xi ∈ Ti(x) for each i ∈ I .

Proof. We may assume each Ei is complete (for the conditions on Ti remain unchanged in the completion of Ei ).Now by applying Michael’s Theorem 1.2 [11], each Ti has a continuous selection fi : X → Xi . Define f : X → X

by f (x) = ∏i∈I fi(x) for each x ∈ X. Then f : X → X is a compact continuous function. By Theorem 4.1, f has a

fixed point x ∈ X; that is, x = f (x) and hence xi = fi(x) ∈ Ti(x) for each i ∈ I . �Theorem 7.2. Let {Xi}i∈I be a family of convex sets, each in a locally convex t.v.s. Ei , Ki a nonempty compactmetrizable subset of Xi , and Ti : X � Ki a Michael map for each i ∈ I . Then there exists an x ∈ K such thatxi ∈ Ti(x) for each i ∈ I .

Proof. Since K := ∏j∈I Kj is compact, coK is a paracompact subset of X as in the proof of Theorem 4.2. Then, by

Michael’s Theorem 1.2 [11], there exists a continuous selection fi : coK → Ei of Ti |coK , where Ei is the completionof Ei , such that fi(x) ∈ Ti(x) ⊂ Ki for all x ∈ coK . Now follow the proof of Theorem 7.1. �Remark. Theorem 7.2 is essentially due to Wu [24, Theorem 1] with much longer proof. Some equilibrium existencetheorems for abstract economies are deduced in [24].

References

[1] H. Ben-El-Mechaiekh, M. Oudadess, Some selection theorems without convexity, J. Math. Anal. Appl. 195 (1995) 614–618.[2] J. Dugundji, A. Granas, Fixed Point Theory, vol. I, Monogr. Math., vol. 61, PWN, Warsaw, 1982.[3] G. Fournier, A. Granas, The Lefschetz fixed point theorem for some classes of non-metrizable spaces, J. Math. Pures Appl. 52 (1973) 271–284.[4] C.J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972) 205–207.[5] C.J. Himmelberg, J.R. Porter, F.S. van Vleck, Fixed point theorems for condensing multifunctions, Proc. Amer. Math. Soc. 23 (1969) 635–641.[6] W. Hurewicz, H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, NJ, 1948.[7] A. Idzik, Almost fixed point theorems, Proc. Amer. Math. Soc. 104 (1988) 779–784.


[8] B. Knaster, K. Kuratowski, S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929) (1988)132–137.

[9] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983) 151–201.[10] E. Michael, Continuous selections. I, Ann. Math. 63 (1956) 361–382.[11] E. Michael, A selection theorem, Proc. Amer. Math. Soc. 17 (1966) 1404–1406.[12] E. Michael, Continuous selections and countable sets, Fund. Math. C XI (1981) 1–10.[13] C.H. Morales, A Bolzano’s theorem in the new millennium, Nonlinear Anal. 51 (2002) 679–691.[14] S. Park, Fixed point theory of multifunctions in topological vector spaces, J. Korean Math. Soc. 29 (1992) 191–208.[15] S. Park, Remarks on fixed points of lower semicontinuous maps, Math. Sci. Res. Hot-Line 2 (3) (1998) 21–26.[16] S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999) 193–232.[17] S. Park, Fixed points of generalized upper hemicontinuous maps revisited, Acta Math. Viet. 27 (2002) 141–150.[18] S. Park, Fixed point theorems in locally G-convex spaces, Nonlinear Anal. 48 (2002) 869–879.[19] S. Park, The KKM principle implies many fixed point theorems, Topology Appl. 135 (2004) 197–206.[20] S. Park, Fixed point theorems for better admissible multimaps on almost convex sets, J. Math. Anal. Appl. 329 (2007) 690–702.[21] S. Park, D.H. Tan, Remarks on Himmelberg–Idzik’s fixed point theorem, Acta Math. Viet. 25 (2000) 285–289.[22] S. Reich, Fixed points in locally convex spaces, Math. Z. 125 (1972) 17–31.[23] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Acad. Publ., Dordrecht, 1998.[24] X. Wu, A new fixed point theorem and its applications, Proc. Amer. Math. Soc. 125 (1997) 1779–1783.



A class of functions whose sublevel sets are absolute retracts

Biagio Ricceri

Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Received 5 October 2006; received in revised form 30 January 2007; accepted 30 January 2007

To Professor Ernest Michael, with esteem, on his eightieth birthday

Abstract

In this paper, we prove a result of which the following is a corollary: If X is a Banach space and J : X → R is a contraction,then the nonempty sublevel sets of the function x → ‖x‖ + J (x) are absolute retracts.© 2007 Elsevier B.V. All rights reserved.

MSC: 54C30; 54C55; 54C60; 54H25

Keywords: Sublevel set; Absolute retract; Set-valued contraction; Fixed point

The notion of an absolute retract is certainly one of the most important in topology.So, let us recall that a nonempty set A in a topological space X is a retract of X if there is a continuous function

g : X → A such that g(x) = x for all x ∈ A. The space X is said to be an absolute retract if, for every metrizablespace Y , any closed subset of Y homeomorphic to X is a retract of Y . Also, recall that any retract of a convex set in anormed space is an absolute retract. Absolute retracts are pathwise connected.

Given a function f : X → R, a sublevel set of f is a set of the type {x ∈ X: f (x) � r}, for some r ∈ R.In the present very short paper, we are interested in presenting a quite general class of functions whose sublevel

sets are absolute retracts.Actually, apart from the case of quasi-convex functions (i.e. functions whose sublevel sets are convex), we did not

find in the literature some other general class of functions with this property.Although, in this paper, our point of view is merely theoretical, we do mention that functions with connected

sublevel sets play an important role in optimization theory as well as in minimax theory (see, for instance, [2,5,7] andthe references therein). We also refer to [1] and [4] for recent, somewhat unexpected applications of retracts of variouskinds.

Before stating our result, let us recall several other notions and the main tool used in the proof.Let E be a real Banach space. The Hausdorff distance between two nonempty sets A, B ⊆ E is defined as

dH (A,B) := max{

supx∈A

infy∈B

‖x − y‖, supy∈B

infx∈A

‖x − y‖}.



872 B. Ricceri / Topology and its Applications 155 (2008) 871–873

Let S ⊆ E and let F : S → 2E be a multifunction with nonempty values. We say that F is Lipschitzian with constantL if

dH

(F(x),F (y)

)� L‖x − y‖

for all x, y ∈ S.Let x ∈ S. We say that x is a fixed point of F if x ∈ F(x). We denote by Fix(F ) the set of all fixed points of F .In [6], making an essential use of the classical Michael selection theorem [3], we obtained the following result:

Theorem A. Let S ⊆ E be a closed and convex set, and let F : S → 2S be a Lipschitzian multifunction with Lipschitzconstant L < 1 and with closed and convex values.

Then Fix(F ) is an absolute retract.

Now, we can state our result.

Theorem 1. Let E be a real Banach space, X ⊆ E a closed and convex set and ϕ : X → R a lower semicontinuousconvex function such that ϕ(x0) = 0 and

α := infx∈X\{x0}

ϕ(x)

‖x − x0‖ > 0

for some x0 ∈ X. Moreover, let J : X → R be a Lipschitzian function with Lipschitz constant L < α.Then each nonempty sublevel set of ϕ + J is an absolute retract.

Proof. Fix r ∈ R in such a way that{x ∈ X: ϕ(x) + J (x) � r

} �= ∅. (1)

Consider the multifunction Φ : R → 2X defined by putting

Φ(t) ={

ϕ−1(]−∞, t]) if t � 0,

{x0} if t < 0.

Let us show that Φ is Lipschitzian in R with Lipschitz constant 1α

. Of course, it suffices to prove this in [0,+∞[. So,let t1, t2 be such that 0 � t1 < t2 and let x ∈ X be such that t1 < ϕ(x) � t2. Put

y =(

1 − t1

t2

)x0 + t1

t2x.

Since ϕ is convex and ϕ(x0) = 0, we then get

ϕ(y) � t1

t2ϕ(x) � t1.

On the other hand,

‖x − y‖ =(

1 − t1

t2

)‖x − x0‖ � t2 − t1

t2

ϕ(x)

α� t2 − t1

α.

This clearly implies that the Hausdorff distance between ϕ−1(]−∞, t1]) and ϕ−1(]−∞, t2]) is less than or equal to(t2 − t1)/α, as claimed. Now, consider the multifunction F : X → 2X defined by putting

F(x) = Φ(r − J (x)

)

for all x ∈ X. So, F is Lipschitzian with Lipschitz constant L/α which is, by assumption, less than 1. Moreover, thevalues of F are closed and convex. We now claim that

{x ∈ X: ϕ(x) + J (x) � r

} = Fix(F ). (2)

The fact that the set on the left-hand side is contained in that on the right-hand side is immediate. We now considerx ∈ Fix(F ). We claim that

J (x) � r. (3)

B. Ricceri / Topology and its Applications 155 (2008) 871–873 873

Arguing by contradiction, suppose that

J (x) > r.

This implies that F(x) = {x0} and hence x = x0. So

J (x0) > r.

Let x ∈ X be a point in the set in (1). Then we would have

α‖x − x0‖ + J (x) � ϕ(x) + J (x) � r < J (x0).

Hence

α‖x − x0‖ < J(x0) − J (x) � L‖x − x0‖.Since ϕ(x0) = 0, we have x �= x0 and so

α < L

contrary to our assumption. Now, from (3), it follows that

F(x) = ϕ−1(]−∞, r − J (x)])

and so x ∈ F(x) means that ϕ(x) + J (x) � r which shows (2). At this point, the conclusion is a direct consequenceof Theorem A. �Remark. It is natural, of course, to ask what happens when L = α. We believe that, in this case, the conclusion ofTheorem 1 is no longer true. But, at present, we have no counter-example.

Acknowledgement

The author wishes to thank the referee for the careful reading of the manuscript.

References

[1] E. Kopecká, S. Reich, Nonexpansive retracts in Banach spaces, Erwin Schrödinger Institute preprint No. 1787, 2006.[2] D.T. Luc, M. Volle, Level sets infimal convolution and level addition, J. Optim. Theory Appl. 94 (1997) 695–714.[3] E. Michael, Continuous selections, I, Ann. of Math. 63 (1956) 361–382.[4] S. Reich, D. Shoikhet, Averages of holomorphic mappings and holomorphic retractions on complex hyperbolic domains, Studia Math. 130

(1998) 231–244.[5] D. Repovs, P.V. Semenov, A minimax theorem for functions with possibly nonconnected intersections of sublevel sets, J. Math. Anal. Appl. 314

(2006) 537–545.[6] B. Ricceri, Une propriété topologique de l’ensemble des points fixes d’une contraction multivoques à valeurs convexes, Rend. Accad. Naz.

Lincei 81 (8) (1987) 283–286.[7] B. Ricceri, On a topological minimax theorem and its applications, in: B. Ricceri, S. Simons (Eds.), Minimax Theory and Applications, Kluwer

Academic Publishers, Dordrecht, 1998, pp. 191–216.



Multivalued maps, selections and dynamical systems

J.J. Sánchez-Gabites 1, J.M.R. Sanjurjo ∗

Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Received 11 October 2006; accepted 12 October 2006

Dedicated to Ernest Michael in the occasion of his 80th anniversary.

Abstract

Under suitable hypotheses the well known notion of first prolongational set J+ gives rise to a multivalued map ψ :X → 2X

which is continuous when the upper semifinite topology is considered in the hyperspace of X. Some important dynamical conceptssuch as stability or attraction can be easily characterized in terms of ψ and moreover, the classical result that an attractor in R

n hasthe shape of a finite polyhedron can be reinforced under the hypotheses that the mapping ψ is small and has a selection.© 2007 Elsevier B.V. All rights reserved.

MSC: 54C60; 54C65; 54H20

Keywords: Multivalued maps; Selections; Upper semifinite topology; Dynamical systems; Attractors

1. Introduction

In his paper [15], E. Michael carried out a thorough study of various topologies on the collection of nonemptyclosed subsets of a topological space X (denoted here CL(X), the reader should keep in mind that this is not thenotation used in [15]) and explored the connection between them and multivalued maps. One of the features of thistheory which is specially motivating for us is that relating the upper semicontinuity of a multivalued map F :X → Y

with its continuity when considered as a map whose target is CL(Y ) endowed with a suitable topology (the so-calledupper semifinite topology) and the subsequent development of the properties of such maps.

The purpose of this article is to take advantage of the theory established by Michael to obtain neat characterizationsof such important notions as stability or attracting sets (which are central in the realm of continuous dynamical sys-tems) in terms of a naturally defined multivalued map ψ obtained from the well known classical tool of prolongationalsets.

Finally, [15] also presents some results about selections of multivalued maps. These will be put into use hereby reinforcing some classical results on the shape of attractors under some hypotheses involving the existence of aselection for ψ .

* Corresponding author.E-mail addresses: [email protected] (J.J. Sánchez-Gabites), [email protected] (J.M.R. Sanjurjo).

1 The authors are partially supported by DGI.


J.J. Sánchez-Gabites, J.M.R. Sanjurjo / Topology and its Applications 155 (2008) 874–882 875

2. Preliminary concepts

We shall devote this section to present the reader with the basic notions regarding hyperspaces, flows and shapetheory while fixing the notation to be used as well as including bibliographical resources for more extensive informa-tion.

All topological spaces are Hausdorff.

2.1. Hyperspaces

Let X be a topological space. A hyperspace of X is a distinguished collection of subsets of X, for example, theone we will be dealing with, which is 2X = {A ⊆ X: ∅ �= A is compact} (this departs from the notation introduced byMichael in [15] but there does not seem to be a universally accepted convention). The hyperspace 2X will be endowedwith the so-called upper semifinite topology, defined as the coarsest possible topology with the property that for anyopen U ⊆ X the set 〈U 〉 = {A ∈ 2X: A ⊆ U} ⊆ 2X is open in 2X . It is easy to check that the sets {〈U 〉: U is openin X} already form a basis for the upper semifinite topology in 2X , because 〈U 〉 ∩ 〈V 〉 = 〈U ∩V 〉. Hence in particulara neighborhood basis for a compact set A ⊆ X is given by {〈U 〉: A ⊆ U , with U open in X}.

Concerning multivalued mappings F :Y → 2X , it is easy to prove that such a mapping is continuous at y ∈ Y ifand only if for every open set X ⊇ U ⊇ F(y) there exists an open neighborhood V of y such that F(y) ⊆ U ∀y ∈ V .Finally, a selection for F is a continuous s :Y → X such that s(y) ∈ F(y) ∀y ∈ Y .

For further references dealing with hyperspaces, multivalued maps and selections we recommend the books [12]and [17] and the papers [15,16,18]. A connection between multivalued maps, selections, coselections and shape theoryhas been studied in [20].

Notation 1. EX(x) (or E(x) if X is understood) will denote the set of open neighborhoods of x in X and similarly,for a compact set A ⊆ X, EX(A) (or E(A) if X is understood) will denote the set of open neighborhoods of A in X.In order to distinguish a compact subset of X from the same object viewed as a point in 2X we will use upper andlower case letters, respectively. Thus A ⊆ X is a compact subset of X while a ∈ 2X is the set A as an element of thehyperspace 2X .

2.2. Flows

A continuous flow in a topological space X (which is usually called the phase space) is an action of the additivegroup R on X or, equivalently, a continuous mapping ϕ :X ×R → X subject to the conditions (1) ϕ(x,0) = x ∀x ∈ X

and (2) ϕ(ϕ(x, t), s) = ϕ(x, t + s) ∀x ∈ X ∀s, t ∈ R. This notion comes from the realm of differential equations, andit still retains some of its original flavor. Thus we usually think of ϕ as a description of the evolution in time t ∈ R ofsome dynamical system with states x ∈ X.

Notation 2. Let us adhere to the customary notation A · I for ϕ(A× I ) when A ⊆ X and I ⊆ R. Conditions (1) and (2)in the definition of flow above can be rewritten as (1) x · 0 = x ∀x ∈ X and (2) x · (t + s) = (x · t) · s ∀x ∈ X ∀s, t ∈ R.

A set K ⊆ X is called positively invariant if K · [0,+∞) = K (thus every point of K stays in K in forwardtime). Similarly, K is negatively invariant if K · (−∞,0] = K and invariant whenever K · R = K (that is, K is bothpositively and negatively invariant).

There exist two basic notions about flows which will be of interest to us, their dynamical significance being quiteprominent. The first one of these is stability, and the second one that of an attracting set.

2.3. Stability

A set K ⊆ X is (positively) stable when points near K stay in its proximity in forward time. More formally, forevery open neighborhood U of K there exists a neighborhood V ⊆ U of K which is positively invariant. A veryuseful tool for studying the stability of sets is that of a prolongational set. Namely, for a point x ∈ X its first (forward)prolongational set is defined in [1] as

876 J.J. Sánchez-Gabites, J.M.R. Sanjurjo / Topology and its Applications 155 (2008) 874–882

J+(x) = {y ∈ X: there exist sequences xn → x and tn → +∞ such that xn · tn → y}and for a set A ⊆ X we put J+(A) = ⋃

x∈A J+(x).For phase spaces X which satisfy the first axiom of numerability, an equivalent definition is J+(x) =⋂

t�0,U∈EX(x) U · [t,+∞) and we will adopt this one in the sequel because it is more general and easier to han-dle in our context. It is clear that J+(x) is closed (though not necessarily nonempty) and an easy argument shows thatit is invariant. These prolongational sets have classically been used to explore the stability and attraction properties ofinvariant sets, as in [1].

2.4. Attractors and repellers

Let K ⊆ X be an invariant compact set. A point x ∈ X is attracted by K if for every open neighborhood U ofK in X there exists some t ∈ R such that x · [t,+∞) ⊆ U (intuitively, x gets closer and closer to K when timegoes forward). An equivalent formulation is obtained by introducing the notion of ω-limit set of a point, defined asω(x) = ⋂

t�0 x · [t,+∞). Then x is attracted by K if and only if ∅ �= ω(x) ⊆ K , at least for locally compact phasespaces. Observe that, similarly to the case of J+(x), the ω-limit set ω(x) is closed and invariant.

The set A(K) = {x ∈ X: x is attracted to K} = {x ∈ X: ∅ �= ω(x) ⊆ K} is called the basin (or region) of attractionof K and clearly contains K . If K is (positively) stable and A(K) is a neighborhood of K , then K is said to be anattractor.

A repeller is the dual concept of an attractor, obtained from the above definitions by reversing time. Thus a pointx ∈ X is repelled by a compact invariant set K∗ if for every open neighborhood U of K∗ in X there exists some t ∈ R

such that x · (−∞, t] ⊆ U . The α-limit set of x is similarly defined as α(x) = ⋂t�0 x · (−∞, t] and, as above, a point

x is repelled by K∗ if and only if ∅ �= α(x) ⊆ K∗, at least for locally compact phase spaces. A compact invariantnegatively stable set K∗ is said to be a repeller if its basin of repulsion R(K∗) = {x ∈ X: x is repelled by K∗} is aneighborhood of K∗.

2.5. Shape theory, internal movability and weak homotopy type in the sense of Borsuk

The fundamentals of shape theory can be found in [6] (or, for a more recent exposition, in [8] and [14]) and wewill not give the relevant definitions since we shall use it only in passing. However let us recall here the notions ofmovability and internal movability.

The first one was already introduced by Borsuk in his article [5] and amounts to the following: a compact space K

(which we can assume without loss of generality to be embedded in some absolute neighborhood retract M , see [4]or [11] for an account of the theory of retracts) is movable if for every U ∈ EM(K) there exists U0 ∈ EM(K) suchthat for every V ∈ EM(K) a continuous mapping H :U0 × [0,1] → U can be found with the property that H(x,0) =x ∀x ∈ U0 and H(x,1) ∈ V ∀x ∈ U0. Thus the identity idU0 can be homotoped, within U , to a continuous mappingwhose image is contained in V .

As for internal movability, it was introduced by Bogatyı in [2] as follows: a compact space K (which, again, weassume embedded in some absolute neighborhood retract M) is internally movable if for every U ∈ EM(K) thereexists U0 ∈ EM(K) and a continuous mapping H :U0 × [0,1] → U such that H(x,0) = x ∀x ∈ U0 and H(x,1) ∈ K

∀x ∈ U0. Thus the identity idU0 can be homotoped, within U , to a continuous mapping whose image is containedin K .

Both movability and internal movability do not depend on the absolute neighborhood retract M where the spacesin question are embedded. Indeed movability is a shape invariant, whereas internal movability is not. However, it wasproved by Dydak [7] that every movable compactum has the shape of an internally movable one, thus its relevance inthe context of shape theory. Further good properties of internally movable compacta can be found in [13].

Finally let us describe the related notion of weak homotopy type in the sense of Borsuk (this terminology is sug-gested by that employed in [2] and following the definition of weakly homotopic maps given in [6, §8, Chapter I,p. 27] but has different meanings in other contexts, so the reader should be aware of this). Two compact spaces K andL embedded in absolute neighborhood retracts M and N have the same weak homotopy type in the sense of Borsukif there exist continuous maps f :K → L and g :L → K such that g ◦ f and f ◦ g are homotopic to the identity inany prescribed neighborhood of K and L, respectively. More precisely, for every U ∈ EM(K) and every V ∈ EN(L)


there exist homotopies g ◦ f � idK (in U ) and f ◦ g � idL (in V ). Again, the choice of M and N turns out to beinmaterial. Having the same weak homotopy type in the sense of Borsuk is weaker than having the same homotopytype but stronger than having the same shape.

3. The continuity of the mapping ψ

Now we shall proceed to define formally the mapping ψ referred to in the introduction. Let a dynamical system begiven on a phase space X which is locally compact and set ψ(x) = J+(x) for every x ∈ X. The set ψ(x) is alwaysclosed, albeit it need not be nonempty. Thus to get a multivalued mapping we have to restrict the domain of ψ .

Notation 3. Define D(ψ) = {x ∈ X: ∅ �= J+(x) is compact}. We shall consider ψ as a multivalued mapping whosedomain is D(ψ).

It turns out that this is the only caveat one has to keep in mind: our first results aim at proving that ψ is continuouswhere defined.

Proposition 4. Let K ⊆ X be compact. For every precompact open neighborhood U of J+(K) there exist t � 0 andV open neighborhood of K such that V · [t,+∞) ⊆ U .

Proof. Step 1. Let us prove first the result when K = {x} is a singleton. Since U is a neighborhood of J+(x), itfollows that ∅ = ∂U ∩ J+(x) = ∂U ∩ ⋂

t�0,W∈EX(x) W · [t,+∞) = ⋂t�0,W∈EX(x)(∂U ∩ W · [t,+∞)). Every ∂U ∩

W · [t,+∞) is compact because it is closed in the compact set ∂U , so by the finite intersection property there mustexist some t � 0 and W ∈ EX(x) such that ∂U ∩ W · [t,+∞) = ∅.

For every w ∈ W one has the following alternative: either w · t ∈ U , in which case w ·[t,+∞) ⊆ U , or w · t ∈ X−U ,in which case w · [t,+∞) ⊆ X − U . Thus the equalities {w ∈ W : w · [t,+∞) ⊆ U} = {w ∈ W : w · t ∈ U} =W ∩U · (−t) and {w ∈ W : w · [t,+∞) ⊆ X−U} = {w ∈ W : w · t ∈ X−U} = W ∩ (X−U) · (−t) hold and thereforethe sets Wi = {w ∈ W : w · [t,+∞) ⊆ U} and Wo = {w ∈ W : w · [t,+∞) ⊆ X − U} are open. If x ∈ Wo, we wouldhave J+(x) ⊆ Wo · [t,+∞) ⊆ X − U , in contradiction with the hypothesis. Choose V = Wi and the proposition isproved for K = {x}.

Step 2. Let us go on to the general case. Applying what we have just seen, for every x ∈ K there exist tx � 0 andVx open neighborhood of x such that Vx · [tx,+∞) ⊆ U . Due to the compactness of K there exist x1, x2, . . . , xn ∈ K

such that K ⊆ ⋃nj=1 Vxj

, so letting t = max1�j�n tj and V = ⋃nj=1 Vxj

(which is clearly an open neighborhoodof K) we get V · [t,+∞) = ⋃n

j=1 Vxj· [t,+∞) ⊆ ⋃n

j=1 Vxj· [tj ,+∞) ⊆ U . �

Proposition 5. Assume X is locally compact. Then ψ :D(ψ) → 2X is continuous. Moreover, the mappingΨ : 2D(ψ) → 2X given by Ψ (K) = ⋃

x∈K ψ(x) = J+(K) is well defined and continuous, too (recall that the topologyin 2X is the upper semifinite topology).

Proof. Let U ∈ EX(J+(x)) be an open neighborhood of J+(x) = ψ(x) and choose a smaller precompact neighbor-hood V of ψ(x), that is J+(x) ⊆ V ⊆ V ⊆ U . Then there exist t � 0 and an open neighborhood W of x such thatW · [t,+∞) ⊆ V . Now for every y ∈ W we have ψ(y) = J+(y) ⊆ W · [t,+∞) ⊆ V ⊆ U . Thus ψ(y) ⊆ U for everyy ∈ W , so ψ :X → 2X is continuous.

We have to check that J+(K) is a compact subset of X to guarantee that Ψ is well defined. But this is one ofthe characteristic features of the upper semi-finite topology (see [15, Corollary 9.6, p. 180]): since the restrictionψ |K :K → 2X is upper semicontinuous and every J+(x) is compact, the union

⋃x∈K J+(x) is also compact. Now

the continuity of Ψ : 2D(ψ) → 2X follows much in the same way as that of ψ . �Let us present a pair of concrete examples in which D(ψ) can be explicitly determined or, at least, sufficiently

precised.

Example 6. Assume that the phase space X is compact. In that case J+(x) is an intersection of a collection ofnonempty closed sets and the finite intersection property implies that J+(x) �= ∅ and (being closed in X) compact,


hence it is an element of 2X and D(ψ) = X. If X is not compact, but locally compact, then one can still constructits one-point (Alexandroff) compactification X∞ and extend the flow to X∞ just by leaving the point at infinity fixed(it is not difficult to check that this is indeed possible). Thus one obtains a compact phase space which essentiallycontains the same dynamics as the original one.

Example 7. Attractors in locally compact phase spaces. Another situation of interest is that of an attractor K in alocally compact space X, since then ∅ �= J+(x) ⊆ K for every x ∈ A(K). This means that J+ is well defined onA(K), or equivalently A(K) ⊆ D(ψ). This will be enough for our purposes since we will be studying the dynamicsjust near K and A(K) is an open invariant neighborhood of it.

4. Some characterizations in terms of ψ

The next theorem characterizes important dynamical properties such as stability and attraction in terms of easyproperties of ψ .

Theorem 8. Let X be a locally compact space where a continuous flow is given such that K ⊆ X is compact andinvariant. Then

(1) K is stable if and only if Ψ (k) = k,(2) K is a global attractor if and only if Ψ ({k}) = k,(3) K is an attractor if and only if Ψ ({k} ∩ W) = k for some neighborhood W of k in 2X .

Proof. (1) The equality Ψ (k) = k means, explicitly, J+(K) = K . Thus what has to be demonstrated is that K isstable if, and only if, J+(K) = K and J+(x) �= ∅ ∀x ∈ K (this guarantees that K ⊆ D(ψ) or equivalently k ∈ 2D(ψ)).

(⇒) Suppose K is stable. For the reader’s convenience we shall prove the inclusion J+(K) ⊆ K although itis already known, see for example [1, Theorem 1.12, p. 61]. Pick x ∈ K and observe that it is enough to see thatJ+(x) ⊆ U for every U ∈ EX(K) because K is compact. So let U ∈ EX(K) and choose a smaller neighborhoodV ∈ EX(K) such that V ⊆ U . Now use the stability of K to obtain a positively invariant W ∈ EX(K) with the propertythat W ⊆ V . Then J+(x) = ⋂

t�0,O∈EX(x) O · [t,+∞) ⊆ W · [t,+∞) ⊆ W ⊆ V ⊆ U , which is what we wanted.

Finally observe that for sufficiently big t � 0 and sufficiently small O ∈ EX(x) the sets O · [t,+∞) are compact, thusJ+(x) �= ∅ by the finite intersection property.

We have seen J+(K) ⊆ K so far and, to prove the equality, let us assume that the inclusion is proper and reasonby contradiction. If J+(K) ⊂ K there would exist some x ∈ K which does not belong to J+(K), but then we couldfind a precompact open neighborhood U of J+(K) such that x /∈ U . By Proposition 4 for some t � 0 we would haveK · [t,+∞) ⊆ U which leads to the contradiction x ∈ K = K · [t,+∞) ⊆ U .

(⇐) The hypothesis now is that J+(K) = K . Let U ∈ EX(K). By Proposition 4 there exist t � 0 and V ∈ EX(K)

such that V · [t,+∞) ⊆ U . Now observe that V · [t,+∞) ∈ EX(K) (because it is the union⋃

s�t V · s and everyV · s ∈ EX(K)) is positively invariant. Thus we have found a positively invariant neighborhood of K contained in U ,which finishes the proof of this first part.

(2) (⇒) Assume that K is a global attractor in X. We must prove that for any p ∈ {k} the equality Ψ (p) = k holds(observe that by Example 7 the domain of Ψ is all of 2X). So let p ∈ {k} and observe that for every U ∈ EX(P ) wehave k ∈ 〈U 〉, or U ⊇ K , which implies P = ⋂

U∈EX(P ) U ⊇ K . Thus K ⊇ J+(P ) ⊇ J+(K) = K , where the firstinclusion is due to the fact that K is an attractor and the last equality owes to the stability of K (see the proof ofpart (1) above). Altogether this gives Ψ (p) = k, as desired.

(⇐) Now our hypothesis is Ψ (p) = k for every p ∈ {k}. In particular Ψ (k) = k, which by (1) above implies thatK is stable. We are left to check that K attracts every point in X. Letting x ∈ X and P = K ∪ {x}, it is clear thatp ∈ {k} so J+(x) ⊆ Ψ (p) = k. Now if U is any open neighborhood of K , by Proposition 4 there exists t � 0 suchthat x · [t,+∞) ⊆ U , so K attracts x. This finishes the proof.

(3) (⇒) We assume that K is an attractor in K . Then A(K) is an open neighborhood of K in X, thus W = 〈A(K)〉is a neighborhood of k in 2X included in the domain of Ψ , by Example 7. Any p ∈ {k} ∩ W satisfies K ⊆ P ⊆ A(K),so again (arguing as in part (2)) J+(P ) = K , that is Ψ (p) = k. This proves Ψ ({k} ∩ W) = {k}.


(⇐) We assume that Ψ ({k} ∩ W) = {k} for some neighborhood W of k in 2X . The same argument as in (2) provesthat K is stable. Now let us show that A(K) is a neighborhood of K . Take U ∈ EX(K) such that k ∈ 〈U 〉 ⊆ W (whichexists because the collection {〈U 〉: K ⊆ U , U is open in X} is a neighborhood basis of k in 2X) and observe that forany x ∈ U the compact set P = K ∪ {x} belongs to {k} ∩ W , hence by hypothesis Ψ (p) = k and J+(x) ⊆ K . Thisimplies that K attracts every point in U , so U ⊆ A(K) which is therefore a neighborhood of K . Consequently K isan attractor. �

When the phase space X is compact and K is an attractor there exists a so-called dual repeller of K , denoted byK∗, which is the set of points not attracted by K , namely K∗ = X − A(K). It can be seen (in [19] for example)that K∗ is indeed a repeller whose basin of repulsion is R(K∗) = X − K . In this situation, another instance of acharacterization in terms of ψ is given by the following proposition.

Theorem 9. Let K ⊆ X be an attractor in a compact phase space X and let K∗ be its dual repeller. Then int(K) = ∅ ifand only if Ψ (k∗) = x ∈ 2X (observe that x ∈ 2X represents the whole phase space X when viewed as a point in 2X).

Proof. We assert that J+(K∗) = R(K∗). The inclusion ⊆ is clear because R(K∗) is an invariant open neighborhoodof K∗. For the reverse one, first we shall prove that for any p ∈ R(K∗) and q ∈ α(p) ⊆ K∗ we have p ∈ J+(q). LetU ∈ EX(q) and t � 0. By means of an argument similar to the one in the proof of Proposition 4 it is easy to prove thatthere exists s � t such that p · (−s) ∈ U , thus p ∈ U · s ⊆ U · [s,+∞) and p ∈ ⋂

U∈EX(q),t�0 U · [t,+∞) = J+(q).

Finally, since J+(K∗) is closed and we have just proved that it contains R(K∗), it follows that J+(K∗) ⊇ R(K∗).It only remains to use the equality int(K) = X − R(K∗) = X − J+(K∗) obtained by combining the paragraph

above and the one preceding the statement of the theorem. �5. A consequence of ψ having a selection

Next we shall confine ourselves to Rn as our phase space, so let us denote by d(x, y) the usual Euclidean distance

between two points x, y ∈ Rn and fix the notations Bε

x and BεK for the open balls of radius ε centered at a point x ∈ R

n

and a compact set K ⊆ Rn, respectively (this last ball is the set of points which are at a distance less than ε of some

point in K).It is known (see, for example, [3,9,10,21,22]) that an attractor in R

n (or, more generally, in an ANR) has the shapeof a finite polyhedron. The proof of this relies on two facts: (a) every suitable (to be specified later) compact positivelyinvariant neighborhood P of K in A(K) has the homotopy type (in particular, the shape) of a finite polyhedron and(b) every inclusion j :K ↪→ P is a shape equivalence. Let us explain these in detail.

(a) In order to clarify what we mean by a suitable positively invariant neighborhood of K we have to introducethe notion of a section of A(K) − K . A compact set Σ ⊆ A(K) − K is such a section if for every x ∈ A(K) − K

its trajectory x · R meets Σ precisely in one point x · Σ , that is x · R ∩ Σ = {x · Σ}, and the mapping x �→ x · Σ iscontinuous. It is known that for every neighborhood U of K there exists a section Σ ⊆ U (a proof of this fact can bequickly furnished via Lyapunov functions but we shall not enter this subject and content ourselves referring the readerto [1]).

Now it is not difficult to see that the set P = Σ · [0,+∞) ∪ K is a compact positively invariant neighborhood ofK with boundary ∂P = Σ . Our interest in it lies in the fact that it has the additional nice property of being a retract ofA(K) because the mapping r :A(K) → P given by

r(x) ={

x · Σ if x /∈ P,

x if x ∈ P,

is easily seen to be continuous and fixes every point in P . Thus P is a compact ANR (since it is a retract of theopen set A(K) ⊆ R

n, which is therefore an ANR) and by a result of West [23] it has the homotopy type of a finitepolyhedron.

It should be observed that if U is a positively invariant neighborhood of K (and these form a neighborhood basisof K , since it is stable), letting Σ ⊆ U be any section of A(K) − K contained in U the construction above yields aneighborhood P which is contained in U . Thus K has a neighborhood basis comprised of compact positively invariantsets which have the homotopy type of a finite polyhedron.


(b) It remains to show that every inclusion j :K ↪→ P is a shape equivalence, where P is a compact positivelyinvariant neighborhood of K (not necessarily of the type specified above). This is done by means of the flow inducedshape morphism from P to K , defined as follows: for any k ∈ N one can consider the homeomorphism rk :A(K) →A(K) defined by rk(x) = x · k, which when restricted to a positively invariant neighborhood P of K gives rise to acontinuous map rk|P :P → P (though not any longer a homeomorphism). It is easy to check that rk|P � rl |P � idP

(in P ) for any k, l ∈ N and rk|K :K → K is homotopic to the identity idK . Further, for every U ∈ ERn(K) there existsk0 such that if k � k0 then im(rk|P ) ⊆ U . Thus {rk: k ∈ N} is an approximative map from P to K in the sense ofBorsuk [6] which is easily seen to be a shape inverse for the inclusion j :K ↪→ P .

Under some circumstances, the argument outlined above can be improved thus:

Theorem 10. Let K ⊆ Rn be an attractor. Assume that ψ :A(K) → 2K is ε-small, where ε > 0 is such that Bε

K ⊆A(K), and that there exists a selection s :U → K for ψ defined on some neighborhood U of K in A(K). Then K isinternally movable and has the weak homotopy type (in the sense of Borsuk) of a finite polyhedron.

Proof. Step 1. For every compact positively invariant neighborhood P of K contained in U , there exists T > 0 suchthat x · T ∈ Bε

s(x) for all x ∈ P .

Proof. Let P be any compact positively invariant neighborhood of K in A(K) which is contained in U . Now forevery t � 0 let Ft = {x ∈ P : x · [t,+∞) ⊆ Bε

s(x)}. We assert that Ft is open in P , which we proceed to prove.Pick any x0 ∈ Ft and observe that J+(x0) ⊆ Bε

s(x0)since s(x0) ∈ J+(x0) and for y ∈ J+(x0) we have d(y, s(x0)) �

diamJ+(x0) < ε. But also x0 · [t,+∞) ⊆ Bεs(x0)

because x0 ∈ Ft , so consequently x0 · [t,+∞) ⊆ x0 · [t,+∞) ∪J+(x0) ⊆ Bε

s(x0). It follows that x0 · [t,+∞) is compact (because it is closed and bounded) and there exists 0 < δ < ε

such that x0 · [t,+∞) ⊆ x0 · [t,+∞) ⊆ Bδs(x0)

.(1) Due to the continuity of the selection s, there exists an open neighborhood V1 of x0 such that d(s(x), s(x0)) <

12 (ε − δ) for every x ∈ V1. Then, if y ∈ Bδ

s(x0)and x ∈ V1, we get d(y, s(x)) � d(y, s(x0)) + d(s(x0), s(x)) < δ +

12 (ε − δ) < ε so y ∈ Bε

s(x). Thus Bδs(x0)

⊆ Bεs(x) for every x ∈ V1.

(2) By Proposition 4, there exist an open neighborhood V2 of x0 and t ′ � t such that V2 · [t ′,+∞) ⊆ Bδs(x0)

.

(3) Since x0 · [t, t ′] ⊆ Bδs(x0)

, the same holds true for every x in some open neighborhood V3 of x0. Thus

V3 · [t, t ′] ⊆ Bδs(x0)

.Combining these three statements we get the following: if x ∈ V1 ∩ V2 ∩ V3, then x · [t,+∞) = x · [t, t ′] ∪ x ·

[t ′,+∞) ⊆ V3 · [t, t ′] ∪ V2 · [t ′,+∞) ⊆ Bδs(x0)

⊆ Bεs(x). Hence x ∈ Ft and (V1 ∩ V2 ∩ V3) ∩ P ⊆ Ft . This finishes the

proof that Ft is open in P . To complete the demonstration of this Step 1, it only remains to observe that Ft ⊆ F ′t if

t � t ′ and P ⊆ ⋃t�0 Ft by Proposition 4, so the compactness of P implies that there exists some T � 0 such that

P ⊆ FT . Thus x · [T ,+∞) ⊆ Bεs(x) for every x ∈ P , and in particular x · T ∈ Bε

s(x) which is what we wanted.

Step 2. For every V ∈ ERn(K) and every compact positively invariant neighborhood P of K contained in U ∩ BεK

there exists k0 ∈ N such that s|P � rk|P in V for every k � k0.

Proof. Let H :P × [0,1] → Rn be given by

H(x, τ) ={

x · (2T τ) if 0 � τ � 12 ,

(x · T )(2 − 2τ) + s(x)(2τ − 1) if 12 � τ � 1.

H is obviously a homotopy connecting idP and s|P . Since P is positively invariant H(P × [0, 12 ]) ⊆ P ⊆ Bε

K .Moreover, both x · T and s(x) belong to Bε

s(x) by construction, thus by the convexity of the balls H({x} × [ 12 ,1]) ⊆

Bεs(x) ⊆ Bε

K and H(P × [ 12 ,1]) ⊆ Bε

K . Hence im(H) ⊆ BεK ⊆ A(K) is a compact subset of A(K), so there exists

some k0 ∈ N such that im(H) · [k0,+∞) ⊆ V . Thus, if we compose H with any map rk :x �→ x · k with k � k0 weget a new homotopy Gk :P × [0,1] → V such that Gk(x,0) = x · k = rk(x) and Gk(x,1) = s(x) · k = (rk ◦ s|P )(x).But the restriction rk|K :K → K is clearly homotopic to the identity, thus rk|P �Gk

rk ◦ s|P = rk|K ◦ s|P � s|P in V .

Step 3. K is internally movable.


Proof. Let W be any open neighborhood of K . Choose a compact positively invariant neighborhood P of K whichis contained in U ∩ Bε

K (so that Step 2 applies) and in W . For V = int(P ) (or any other open neighborhood of K

contained in P ) there exists k0 ∈ N such that s|P � rk0 |P in V , thus in P . Finally rk0 |P � idP in P , so s|P � idP in P

(and therefore in W too). Hence K is internally movable.

Step 4. K has the weak homotopy type (in the sense of Borsuk) of a finite polyhedron.

Proof. Let P be a positively invariant compact neighborhood of K sufficiently small such that the assertion in Step 2holds true and with the homotopy type of a finite polyhedron (its existence was established prior to the formulation ofthis theorem). It will suffice to show now that K has the weak homotopy type (in the sense of Borsuk) of P . Denotej :K ↪→ P the inclusion.

s|P ◦ j � idK in any neighborhood V of K . To see this, by Step 2 there exists some k0 ∈ N such that s|P � rk0 |Pin V . Now s|P ◦ j � rk0 |P ◦ j = rk0 |K in V . Finally rk0 |K � idK (in K ⊆ V ), so we get s|P ◦ j � idK in V .

j ◦ s|P � idP . This is similar to Step 3, and proceeds as follows. Let V be any open neighborhood of K containedin P , by Step 2 there exists some k0 ∈ N such that s|P � rk0 |P in V , thus in P . But rk0 |P � idP (in P ), so s|P � idP

in P . �Remark 11. An appropriate version of the result above holds true for phase spaces other than R

n. Namely, let X bea locally convex and locally compact metrizable part of a topological vector space (for example, Hilbert’s cube) andassume K ⊆ X is an attractor. For every y ∈ K pick some Uy ∈ EX(y) which is convex, precompact and contained inA(K) (this exists because X is locally convex and locally compact), thus obtaining an open covering U = {Uy : y ∈ K}of K by open sets, and let ε > 0 be a Lebesgue number for U . Then, if ψ :A(K) → 2K is ε-small and has a selections :U → K defined on some neighborhood U of K in A(K), K is internally movable and has the weak homotopy type(in the sense of Borsuk) of a finite polyhedron.

We shall indicate briefly the necessary modifications necessary to the proof of Theorem 10. By the choice of ε

it holds true that for every x ∈ K there exists some Uy such that Bεx ⊆ Uy . With this in mind Step 1 goes through

unchanged, and in Step 2 it suffices to observe that for any x ∈ P we have that both x · T and s(x) belong to Bεs(x),

thus to some convex Uy , so that H({x} × [ 12 ,1]) ⊆ Uy ⊆ A(K). This makes it possible to read along the rest of the

argument without any further changes, only in Step 4 it should be observed that X is an ANR because it is locallyconvex (see [11, Corollary 14.2, p. 60 and Theorem 8.1, p. 98] for a proof).

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157–196.[17] D. Repovs, P.V. Semënov, Continuous Selections of Multivalued Mappings, Mathematics and its Applications, vol. 455, Kluwer Academic

Publishers, 1998.


[18] D. Repovs, P.V. Semënov, Continuous Selections of Multivalued Mappings, Recent Progress in General Topology, II, North-Holland, 2002.[19] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1) (1985) 1–41.[20] J.M.R. Sanjurjo, Selections of multivalued maps and shape domination, Math. Proc. Cambridge Philos. Soc. 107 (3) (1990) 493–499.[21] J.M.R. Sanjurjo, Multihomotopy, Cech spaces of loops and shape groups, Proc. London Math. Soc. (3) 69 (2) (1994) 330–344.[22] J.M.R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl. 192 (2) (1995) 519–528.[23] J.E. West, Mapping Hilbert cube manifolds to ANR’s: a solution of a conjecture of Borsuk, Ann. of Math. (2) 106 (1977) 1–18.



Moscow spaces and selection theory ✩

M. Sanchis

Departament de Matemàtiques, Universitat Jaume I, Campus del Riu Sec. s/n, 12071 Castelló, Spain

Received 2 October 2006; received in revised form 2 February 2007; accepted 2 February 2007

Abstract

A well-known result on Moscow spaces states that every Gδ-dense subset of a Moscow space X is C-embedded in X. We presenthere the selection version of this result and also (by means of two different approaches) we use selection theory to characterize theopen bounded subsets of a uniform space (X,U) in the case when its completion is a Moscow space.© 2007 Elsevier B.V. All rights reserved.

MSC: primary 54C65; secondary 54C45, 54G99

Keywords: Continuous carrier; Lower semicontinuous carrier; Moscow space; u-selection; Bounded subset; Gδ-dense subset; C-embedded subset

1. Introduction

Not long after Michael published in the fifties his papers on selection theory, this subject became a productive fieldof research: selection theory not only generalizes most of the familiar extension theorems but it also permits to obtaininteresting applications. These aspects were already pointed out by Michael in [11–13].

This paper deals with selection theory in the realm of Moscow spaces. Our goal is twofold. Firstly we give theversion in selection theory of one of the most interesting properties of Moscow spaces: every Gδ-dense subset of aMoscow space X is C-embedded in X. Secondly, we characterize the open bounded subsets of a uniform space (X,U)

in the case when its completion is a Moscow space. This situation includes some relevant special cases, for instance,the case of locally precompact groups. (For results in selection theory related to topological groups the reader mightconsult [15].)

Throughout this paper all spaces are assumed to be Tychonoff. If Y is a topological space, 2Y stands for thefamily of nonempty subsets of Y . A function φ :X → 2Y is called a carrier. A selection for a carrier φ :X → 2Y is acontinuous function f :X → Y such that f (x) ∈ φ(x) for every x ∈ X. We deal here with continuous carriers. Let usrecall the definition. If Y is a metric space, a carrier φ :X → 2Y is said to be continuous if, given ε > 0, every x0 ∈ X

has a neighborhood U such that, for every x ∈ U ,

φ(x0) ⊂ Bε

(φ(x)

), φ(x) ⊂ Bε

(φ(x0)

),

where Bε(A) (with A ⊂ Y ) denotes {y ∈ Y | d(y,A) < ε}.

✩ This research is supported by BFM2003-02302 and by Fundació Bancaixa.E-mail address: [email protected].


884 M. Sanchis / Topology and its Applications 155 (2008) 883–888

It is clear that a carrier is continuous if and only if it is continuous with respect to the topology induced by theHausdorff–Bourbaki uniformity Ud on 2Y associated to the metric d . The restriction of Ud to the family C(Y ) of 2Y

formed by all the nonempty closed subsets of Y is Hausdorff and, since it has a countable base, can be generated bya metric. In [9] Kuratowski proved that the restriction of the Hausdorff–Bourbaki uniformity to the family C(Y ) iscomplete and he credited this result to Hahn [10].

We now turn to Moscow spaces. A regular closed set of a space X is the closure of an open set of X. A space X iscalled Moscow [1,2] if every regular closed subset of X is the union of a family of Gδ-subsets of X. (As usual, hereGδ-set means a set which is a countable intersection of open sets.)

Spaces of countable pseudocharacter and extremally disconnected spaces are Moscow. The notion of Moscowspace also generalizes the notion of a perfectly κ-normal space; in particular, locally precompact topological groupsare Moscow (see [16] and [14] for further information and references). More large classes of Moscow spaces arepresent in [5].

This paper is organized as follows. The second section is devoted to see how the above-mentioned theorem onGδ-dense subsets of Moscow spaces can be stated in the realm of selection theory. In the third and fourth sections wetackle the case of open bounded subsets of uniform spaces. We freely use all conventional notation and terminology.For instance, clX A stands for the closure of A in X, f |A for the restriction of a function f to a subset A and R for thereals endowed with the usual topology. For notation and terminology not defined here the reader might consult [8].

2. Gδ-dense subsets of Moscow spaces

Moscow spaces have interesting applications on several fields. One of the most relevant is the field of topologicalgroup (see, for instance, [3,4]). In some of these applications it plays an important role the fact that every Gδ-densesubset S of X is C-embedded in X, i.e., every real-valued continuous function on S can be continuously extended tothe whole X. (Recall that a subset S of a space X is said to be Gδ-dense (in X) if every nonempty Gδ-subset of X

meets Y .)This result was credited to Tkacenko by Uspenskij in [17]. We wish now to consider the selection version of this

theorem. First we need a lemma. Throughout this paper, unless the contrary is stated, Y denotes a metric space (Y, d).We say that a family {Fα}α∈D has the countable intersection property if

⋂n<ω Fαn �= ∅ for every countable subset

{αn}n<ω of D.

Lemma 2.1. Let S be a Gδ-dense subset of a Moscow space X and let φ :X → C(Y ) be a continuous carrier. Iff :S → Y is a continuous selection for φ|S and x0 ∈ X \ S is a cluster point of a filter F on S, then the family

{(clY f (F )

) ∩ φ(x0) | F ∈ F}

has the countable intersection property.

Proof. Let {Fn}n<ω be a countable subfamily of F . Since F is a filter we can assume with no loss of generality thatFn+1 ⊂ Fn for every n < ω. Now, since f is continuous, the sets

An ={s ∈ S | d(

f (s),φ(x0))<

1

n

}, n < ω

and

Bn ={s ∈ S | d(

f (s), clY f (Fn))<

1

n

}, n < ω

are open in S. Moreover, since x0 is a cluster point of the filter F , it is easy to show that x0 ∈ clX An ∩ clX Bn forevery n < ω. Since X is a Moscow space there exists a Gδ-set Gn ⊂ X such that

x0 ∈ Gn ⊂ clX An ∩ clX Bn, n < ω.

Being S Gδ-dense in X, there is s ∈ (⋂

n<ω Gn) ∩ S, that is, s belongs to⋂

n<ω(clS An ∩ clS Bn). Therefored(f (s),φ(x0)) = 0 and d(f (s), clY f (Fn)) = 0 for every n < ω. Thus, s ∈ φ(x0) ∩ (

⋂n<ω clY f (Fn)). �

M. Sanchis / Topology and its Applications 155 (2008) 883–888 885

Let S(Y ) denote the family of elements of C(Y ) which are separable and let K(Y ) ⊆ C(Y ) be the family of allnonempty compact subsets of Y . If B is a base for a (ultra)filter, GB will stand for the (ultra)filter generated by B.We are ready now to prove the following

Theorem 2.1. If S is a Gδ-dense subset of a Moscow space X, then the following assertions hold:

(a) For each metric space Y , if φ :X → S(Y ) is a continuous carrier, then every selection for φ|S can be extended toa selection for φ.

(b) For each metric space Y , if φ :X → K(Y ) is a continuous carrier, then every selection for φ|S can be extendedto a selection for φ.

(c) If φ :X → C(R) is a continuous carrier, then every selection for φ|S can be extended to a selection for φ.(d) If φ :X → K(R) is a continuous carrier, then every selection for φ|S can be extended to a selection for φ.

Proof. It is clear that (b) ⇒ (d). Moreover (a) ⇒ (b) because every compact metric space is separable, and (a) ⇒ (c)because R is a separable metric space. So we only need to prove clause (a). For this let f be a selection for φ|S . Letx ∈ X \S and consider an ultrafilter Fx on S converging to x. By Lemma 2.1, the family {(clY f (F ))∩φ(x) | F ∈ Fx}has the countable intersection property. Since φ(x) is a separable metric space, it is Lindelöf and, consequently, thereis yx ∈ Y such that

yx ∈( ⋂

F∈Fx

clY f (F )

)∩ φ(x).

Notice that yx is a cluster point of the ultrafilter Gf (Fx) which implies that Gf (Fx) converges to yx . We finish the proofby showing that the function f :X → Y defined as

f (x) ={

yx if x ∈ X \ S;f (x) if x ∈ S

is continuous. By [8, 6H] it suffices to prove that f |S∪{x} is a continuous function for every x ∈ X \ S. To see thisconsider x ∈ X \ S and suppose that there exists a filter F on S converging to x such that G

f (F )does not converge to

yx . Choose ε > 0 such that clY Bε(yx) does not contain any element of Gf (F )

. Now consider the open sets (in S)

A = f −1(Bε/2(yx))

and

B = f −1(Y \ clY Bε(yx)).

Since every F ∈ F meets B and A ∈ Fx , we have

x ∈ clX A ∩ clX B.

Now since X is a Moscow space, there is a Gδ-set G in X satisfying x ∈ G ⊂ clX A ∩ clX B . Being S Gδ-dense in X,G ∩ clX A ∩ clX B ∩ S �= ∅ which is a contradiction. Thus, f is continuous. �

As a straightforward consequence of the previous theorem we have

Corollary 2.1 (Tkacenko’s theorem). Every Gδ-dense subset of a Moscow space X is C-embedded in X.

3. Bounded subsets in Moscow spaces

In Sections 3 and 4 (X, U) stands for the completion of a uniform space (X,U). If X is a topological space,a uniformity U on X is called admissible if the topology induced by U coincides with the given topology on X. Thetopological space induced by (X,U) will be denoted by X. When treating uniform spaces we follow notation andterminology adopted in [7] and [8].


We need now to introduce some concepts related to topological spaces and uniform spaces. A subset S of a spaceX is said to be bounded (in X) if every continuous real-valued function on X is bounded on S. So pseudocompactspaces are spaces which are bounded in themselves. A subset A of a uniform space (X,U) is called precompact iffor every entourage V ∈ U of the diagonal there exists a finite subset {x1, x2, . . . , xn} ⊂ X such that A ⊆ ⋃n

i=1 V (xi)

(as usual, V (x) (x ∈ X) stands for the set {y ∈ X | (x, y) ∈ V }). If X is precompact in itself, then X is called aprecompact uniform space. A standard result on uniform spaces states that a uniform space is precompact if and onlyif its completion is compact.

If X is a given topological space, there exists a finest admissible uniformity on X, the so-called universal uni-formity. A space X is said to be topologically complete or Dieudonné complete if the universal uniformity on X iscomplete. For every space X there exists a unique topologically complete space μX, up to homeomorphisms whichleave X pointwise fixed, in which X is dense and every continuous function from X into a topologically completespace can be extended to a continuous function on μX; μX is called the Dieudonné (topological) completion of X.A well-known result tells us that the closure of every bounded set in a topologically complete space is compact. Thenext proposition gives us a characterization of a bounded subset which maybe has interest in itself.

Proposition 3.1. The following are equivalent for a subset S of a space X:

(1) S is bounded in X.(2) For every compatible uniformity U on X, S is precompact in (X,U).(3) S is precompact in the Dieudonné completion μX of X.

Proof. (2) ⇒ (3) is clear. To see (1) ⇒ (2) notice that since (X, U) is complete, the space X is topologically complete.Hence the bounded subset clX S is compact and, consequently, S is precompact in (X,U).

Now we shall show (3) ⇒ (1). For a real-valued continuous function f on X, consider the (continuous) pseudo-metric Ψf defined as Ψf (x, y) = |f (x) − f (y)| for every x, y ∈ X. Since the universal uniform structure on X isgenerated by the set of all continuous pseudometrics on X × X [8, 15G(4)] and S is precompact in μX, we can find afinite subset {x1, x2, . . . , xn} ⊂ X such that

S ⊂n⋃

i=1

{x ∈ X | ∣∣f (xi) − f (x)

∣∣ < 1}

which implies that the restriction of f to S is a bounded function. �Comfort and Ross proved in [6] that every pseudocompact group G is precompact for the left uniformity of G

(equivalently, for the right uniformity of G). This result was generalized by Tkacenko [18] by showing that everybounded subset of a topological group G is precompact for the left (and also for the right) uniformity of G. Proposi-tion 3.1 permits us to generalize these results:

Corollary 3.1. Every bounded subset of a topological group G is precompact for any compatible uniformity of G. Inparticular, if G is a pseudocompact group, then G is precompact for any compatible uniformity of G.

We turn now to bounded subsets in (uniform) Moscow spaces. In general, if a complete uniform space (X, U) hasa dense space which is Moscow, then (X, U) can fail to be Moscow: the one-point compactification of an uncountablediscrete space can serve as the simplest standard example. Thus, we will work in the sequel with uniform spaces whosecompletion is a Moscow space. Notice that if S is an open bounded subset of a space X, then clX S is a pseudocompactspace.

Theorem 3.1. If (X, U) is a Moscow space, then the following conditions are equivalent for an open precompactsubset S of (X,U):

(a) S is bounded in X.(b) For each metric space Y , if φ : clX S → S(Y ) is a continuous carrier, then every selection for φ|clX S can be

extended to a selection for φ.(c) If φ : clX S → C(R) is a continuous carrier, then every selection for φ|clX S can be extended to a selection for φ.

M. Sanchis / Topology and its Applications 155 (2008) 883–888 887

Proof. It suffices to prove (a) ⇒ (b) and (c) ⇒ (a).(a) ⇒ (b) Since S is an open bounded set, clX S is pseudocompact. So clX S is Gδ-dense in the Moscow space

clX S. The result now follows from condition (1) in Theorem 2.1.(c) ⇒ (a) If f is a continuous real-valued function on clX S, f is a selection for φ|clX S where φ is the continuous

carrier φ : clX S → C(R) defined as φ(x) = R for every x ∈ clX S. So clause (c) tell us that f admits a continuousextension to clX S, that is, clX S is C-embedded in clX S. Since S is precompact, clX S is compact. This proves thatclX S is pseudocompact. �

An argument similar to the one in Theorem 3.1 shows

Theorem 3.2. If (X, U) is a compact Moscow space, then the following are equivalent:

(a) (X,U) is pseudocompact.(b) For each metric space Y , if φ : X → S(Y ) is a continuous carrier, then every selection for φ|X can be extended to

a selection for φ.(c) If φ : X → C(R) is a continuous carrier, then every selection for φ|X can be extended to a selection for φ.

4. A different approach

Theorem 3.1′′ in [11] states that (in the realm of T1-spaces) normal countably paracompact spaces are characterizedby the existence of a selection for l.s.c. carries with values in the family of all (nonempty) closed convex subsets of aseparable Banach space. So, it seems that this result cannot be applied to open bounded sets because in general theyare far from being either normal or countably paracompact. However, in Moscow spaces it is possible to apply theabove mentioned Michael’s theorem in order to characterize open bounded subsets. The goal of this section is to pointout how to do this.

Firstly let us introduce some notation and terminology. Given a normed space (E,‖ · ‖), F (E) denotes the familyof nonempty, closed, convex subsets of E. If Y is a topological space, a carrier φ :X → 2Y is lower semicontinuous(or l.s.c., for short) if {x ∈ X | φ(x)∩V �= ∅} is open in X for every open subset V ⊂ Y . In the case when Y is a metricspace, it is an easy matter to show that every continuous carrier φ :X → 2Y is lower semicontinuous.

In [13, Section 8], Michael points out that the existence of selections for continuous carries cannot be used tocharacterize topological properties (in a similar way as it is done in [11] by means of l.s.c. carriers). In fact, Theo-rem 8.1 in [13] states that, if X is a topological space, and Y a Banach space, then every continuous carrier φ :X →F (Y ) admits a selection. However, the following theorem tells us that the situation is quite different if we considerselections with some (natural) additional properties. It is worth mentioning that our result is an application of Theo-rem 3.1′′ in [11]: such an application is possible thanks to Corollary 2.1. But first, a definition.

Definition 4.1. Let (X,U) be a uniform space, and let Y be a normed space. A selection f for a carrier φ :X → 2Y issaid to be a u-selection if f is uniformly continuous.

Theorem 4.1. If (X, U) is a Moscow space, then the following are equivalent for an open precompact subset S of(X,U):

(1) S is bounded.(2) If Y is a separable Banach space, then every continuous carrier φ : clX S → F (Y ) admits a u-selection.(3) Every continuous carrier φ : clX S → F (R) admits a u-selection.

Proof. (1) ⇒ (2) Since the space clX S is pseudocompact, it is precompact in (X,U) (Theorem 3.1) and Gδ-densein the Moscow space clX S. So Corollary 2.1 tells us that clX S coincides with the Stone–Cech compactification ofclX S. Since φ(clX S) is a pseudocompact subset of the metric space F (Y ), it is compact and, consequently, φ admitsa continuous extension φ to clX S. Now, by Theorem 3.1′′ in [11], there exists a selection f : clX S → Y for φ. It isclear that f |clX S is a u-selection for φ.

(2) ⇒ (3) It is clear.


(3) ⇒ (1) For a real-valued continuous function f on clX S, consider the carrier φ : clX S → F (R) defined asφ(x) = {f (x)} for every x ∈ clX S. It is straightforward that φ is continuous. By clause (3) f is a uniformly continuousfunction and, consequently, it admits a uniformly continuous extension to the completion of (clX S,U|clX S), thecompact uniform space (clX S, U|clX S). This proves that clX S is pseudocompact which implies that S is bounded(in X). �

In the same way we can obtain the following

Theorem 4.2. If (X, U) is a Moscow space, then the following are equivalent:

(1) (X,U) is pseudocompact.(2) If Y is a separable Banach space, then every continuous carrier φ :X → F (Y ) admits a u-selection.(3) Every continuous carrier φ :X → F (R) admits a u-selection.

We finish the section with a result that sheds some light on the properties of bounded subsets of topological groups.The proof is a consequence of Theorem 4.1.

Corollary 4.1. Let (X, U) be a Moscow space. If S is an open bounded subset in (X,U), then every continuousfunction f : clX S → R is uniformly continuous.

Remark 4.1. Since locally precompact groups are Moscow, the results obtained in this paper can be applied to therealm of topological groups. In particular, the results presented here generalize some results proved in [15].

Acknowledgements

The author thanks the referee for his/her accurate and useful suggestions.

References

[1] A.V. Arhangel’skii, Functional tightness, Q-spaces, and τ -embeddings, Comment. Math. Univ. Carolin. 24 (1983) 105–120.[2] A.V. Arhangel’skii, Topological homogeneity, topological groups and their continuous images, Russian Math. Surveys 42 (1987) 83–131.

Russian original in: Uspekhy Mat. Nauk 42 (1987) 69–105.[3] A.V. Arhangel’skii, Moscow spaces, Pestov–Tkacenko problem, and C-embeddings, Comment. Math. Univ. Carolinae 41 (3) (2000) 585–595.[4] A.V. Arhangel’skii, Topological groups and C-embeddings, Topology Appl. 115 (3) (2001) 265–289.[5] A.V. Arhangel’skii, On power homogeneous spaces, Topology Appl. 122 (2002) 15–33.[6] W.W. Comfort, K.A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966) 483–496.[7] R. Engelking, General Topology, PWN, Warszawa, 1977.[8] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976. Reprint of Graduate Texts in Math., vol. 43, 1960.[9] C. Kuratowski, Topologie I, Espaces métrisable, Espaces complets. Monografie Mat., Warsaw, 1933.

[10] H. Hahn, Reele Funktionen, Akademische Verlasgesellschaft, Leipzig, 1932.[11] E. Michael, Continuous selections, I, Ann. of Math. 63 (2) (1956) 361–382.[12] E. Michael, Continuous selections, II, Ann. of Math. 64 (1956) 562–580.[13] E. Michael, Continuous selections, III, Ann. of Math. 65 (2) (1957) 375–390.[14] H. Ohta, M. Sakai, K.-I. Tamano, Perfect κ-normality in product spaces, in: Papers on General Topology and Applications, in: Annals of the

New York Academy of Sciences, vol. 704, 1993, pp. 279–289.[15] M. Sanchis, Continuous selections and locally pseudocompact groups, Set-Valued Anal. 12 (2004) 319–328.[16] E.V. Šcepin, On κ-metrizable spaces, Math. USSR Izv. 14 (1980) 407–440. Russian original in: Izv. AN SSSR Ser. Mat. 43 (1980) 442–478.[17] V.V. Uspenskij, Topological groups and Dugundji compacta, Math. USSR Sb. 67 (1990) 555–580. Russian original in: Mat. Sb. 180 (1989)

1092–1118.[18] M.G. Tkacenko, Compactness type properties in topological groups, Czech. Math. J. 38 (1988) 324–341.



Collectionwise normality and selections into Hilbert spaces

Ivailo Shishkov

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev Str. bl. 8, 1113 Sofia, Bulgaria

Received 22 August 2006; received in revised form 30 April 2007; accepted 30 April 2007

Abstract

A T1-space X is countably paracompact and collectionwise normal if and only if every l.s.c. mapping from X into a Hilbertspace with closed and convex point-images has a continuous selection. This settles a conjecture posed by M. Choban, V. Gutev andS. Nedev [M. Choban, S. Nedev, Continuous selections for mappings with generalized ordered domain, Math. Balkanica (N.S.) 11(1–2) (1997) 87–95].© 2007 Elsevier B.V. All rights reserved.

MSC: 46C05; 46A25; 54C60; 54C65; 54D15

Keywords: Set-valued mapping; Lower semi-continuous; Selection; Hilbert space

1. Introduction

The famous Michael selection theorem from [4] asserts the following characterization of paracompactness.

Theorem 1.1. (See [4, Theorem 3.2′′].) For a T1-space X, the following conditions are equivalent:

(a) X is paracompact;(b) whenever Y is a Banach space, every l.s.c. mapping Φ :X → Fc(Y ) has a continuous selection.

Here, F(Y ) is the set of all nonempty closed subsets of Y , and Fc(Y )—that of all convex members of F(Y ).A mapping Φ from X into the nonempty subsets of Y is lower semi-continuous, or l.s.c., if Φ−1(U) = {x ∈ X: Φ(x)∩U �= ∅} is open in X for every open U ⊂ Y . A map f :X → Y is a selection for Φ if f (x) ∈ Φ(x) for every x ∈ X.

Concerning Theorem 1.1, let us explicitly mention that, in fact, Michael has shown a little bit more in [4] that aT1-space X is paracompact if, whenever U is an open cover of X, every l.s.c. mapping Φ :X → Fc(�1(U)) has acontinuous selection. Recall that �1(U) is the Banach space of all functions y :U → R, with

∑{|y(U)|: U ∈ U} < ∞,equipped with the norm ‖y‖ = ∑{|y(U)|: U ∈ U}.

On the other hand, Theorem 1.1 fails if “Banach space” is replaced by “Hilbert space”. It was demonstrated in [8]that every l.s.c. mapping Φ :ω1 → Fc(Y ), where Y is a Hilbert space and ω1 is the first uncountable ordinal, has acontinuous selection. Later on, this result was generalized in [1] by replacing ω1 with an arbitrary generalized orderedspace. It is well known that every generalized ordered space is countably paracompact and collectionwise normal,but not necessarily paracompact (see, e.g., [2]). Finally, let us also mention that a T1-space X must be countably


890 I. Shishkov / Topology and its Applications 155 (2008) 889–897

paracompact and collectionwise normal provided every l.s.c. mapping Φ :X → Fc(Y ), where Y is a Hilbert space,has a continuous selection

Motivated by this, the following question was raised by M. Choban, V. Gutev and S. Nedev.

Question 1.1. (See [1].) Let X be a collectionwise normal and countably paracompact space, Y be a Hilbert space,and let Φ :X → Fc(Y ) be l.s.c. Does Φ admit a continuous selection?

Several partial cases of Question 1.1 were resolved, see, for instance, [9–12]. In particular, the following reductionof Question 1.1 to the case of special l.s.c. mappings was obtained.

Lemma 1.2. (See [9].) Let Y be a normed space, and X be a countably paracompact space. Then every l.s.c. mappingΦ :X → Fc(Y ) has a continuous selection if (and only if ) every bounded l.s.c. mapping Ψ :X → Fc(Y ) has acontinuous selection.

Here, a mapping Ψ :X →F(Y ) is bounded if there exists a positive number r such that Ψ (x) ⊂ {y ∈ Y : ‖y‖ � r}for all x ∈ X, where ‖.‖ is the norm of Y .

The goal of this paper is to prove the following theorem which, by virtue of Lemma 1.2, provides the completeaffirmative solution to Question 1.1.

Theorem 1.3. Let X be a collectionwise normal space, Y be a Hilbert space, and let Φ :X → Fc(Y ) be an l.s.c.mapping such that each Φ(x), x ∈ X, is bounded. Then, Φ has a continuous selection.

The proof of Theorem 1.3 will be finalized in Section 3. Briefly, Section 2 deals with some geometric propertiesof Hilbert spaces, while Section 3 collects technical results which allow to apply Michael’s method (see [4, Proof ofTheorem 3.2′′]) of constructing of selections.

2. Some geometric properties of Hilbert spaces

We use all conventional notations such as N, Q and R to denote, respectively, the set of natural numbers, the setof rational numbers and that one of real numbers. If Y is a normed space endowed with a norm ‖.‖, we will use B todenote the unit open ball {y ∈ Y : ‖y‖ < 1}, and we let Bε = ε · B = {y ∈ Y : ‖y‖ < ε}, ε > 0. Also, if y ∈ Y , then,for convenience, we let Bε(y) = y + Bε = {z ∈ Y : ‖y − z‖ < ε}. Finally, if F ⊂ Y , then conv(F ) and conv(F ) willdenote, respectively, the convex hull of F and the closed convex hull of F .

By a cardinal we mean an initial ordinal. Given a cardinal number τ � ω, where ω is the first infinite ordinal,we will use H to denote the Hilbert space �2(τ ) of all functions f : τ → R, with

∑{f 2(α): α < τ } < ∞, endowedwith the standard dot product 〈f,g〉 = ∑{f (α) · g(α): α < τ }, f,g ∈ H. Also, let ‖f ‖ = √〈f,f 〉, f ∈ H, be thecorresponding norm on H generated by the dot product.

Lemma 2.1. For every i ∈ N, let mi ∈ H be such that mi(α) � mi(α + 1) � 0 for all α < τ . Also, let c > 0 be suchthat

∑α<τ mi(α) � c for every i ∈ N. Then {mi}∞i=1 has a convergent subsequence.

Proof. Note that the sequence {mi}∞i=1 is bounded. Indeed, for any i ∈ N, we have that

‖mi‖2 = 〈mi,mi〉 =∑

α<τ

(mi(α)

)2 �∑

α<τ

c · mi(α) = c ·∑

α<τ

mi(α) � c2.

Since every reflexive Banach space is locally weakly sequentially compact (see, for instance, [13]), there is a weaklyconvergent subsequence {mij }∞j=1 of {mi}∞i=1. For convenience, let yj = mij , j ∈ N, and let y be the limit of {yj }∞j=1in the weak topology of H. It now suffices to show that limj→∞ ‖y − yj‖ = 0.

I. Shishkov / Topology and its Applications 155 (2008) 889–897 891

Turning to this purpose, let us observe that for every δ > 0 there exists a k(δ) < ω � τ such that mi(k(δ)) < δ forall i ∈ N. Namely, conversely to what we claim, suppose that there exists a δ > 0 such that for every k < ω there is ani(k) ∈ N, with mi(k)(k) � δ. Then, whenever k � c/δ, we have

∑

α<τ

mi(k)(α) �k∑

j=0

mi(k)(j) � (k + 1)δ � (c/δ + 1)δ > c,

which contradicts the hypotheses of our lemma.Now let ε > 0, and let k = k(ε2/4c) be as above. Since the weak convergence of {yj }∞j=1 means that

limj→∞ yj (α) = y(α) for each α < τ , there is an n(ε) ∈ N such that∑

α�k(y(α) − yj (α))2 < ε2/2 for every

j > n(ε). On the other hand, 0 � y(α) � ε2/4c for every α > k because, in this case, α > k implies that0 � yj (α) � yj (k) < ε2/4c, j ∈ N. Finally, take in mind that

∑α<τ yj (α) � c, j ∈ N, implies

∑α<τ y(α) � c.

Therefore,

∑

α>k

(y(α) − yj (α)

)2 �∑

α>k

∣∣y(α) − yj (α)∣∣ · ε2

4c� ε2

4c·∑

α>k

(y(α) + yj (α)

)� ε2 · 2c

4c= ε2

2.

Hence, for every j > n(ε), we have

‖y − yj‖2 =∑

α�k

(y(α) − yj (α)

)2 +∑

α>k

(y(α) − yj (α)

)2<

ε2

2+ ε2

2= ε2,

which completes the proof. �For every y ∈ H, let supp(y) = {α < τ : y(α) �= 0}. The following special subsets of H will play a crucial role in

our next considerations.

Hfin = {

y ∈ H:∣∣supp(y)

∣∣ < ∞},

Hfin+ = {

y ∈ Hfin: y(α) � 0 for all α < τ

}, and

Hfin− = {

y ∈ Hfin: y(α) � 0 for all α < τ

}.

Let Bı(τ ) stand for the set of all bijections of τ onto itself. To every ϕ ∈ Bı(τ ) assign ϕ : H → H by the formula

ϕ(y)(α) = y(ϕ(α)

), α < τ.

According to this definition, we get the following immediate properties of ϕ.

Proposition 2.2. If ϕ ∈ Bı(τ ), then

(i) ϕ is a linear bijection,(ii) (ϕ)−1 = ϕ−1,

(iii) 〈ϕ(x),ϕ(y)〉 = 〈x, y〉, x, y ∈ H.

Finally, for any y ∈ H, we consider also the following subsets of H.

Orbit(y) = {ϕ(y): ϕ ∈ Bı(τ )

},

P(y) = {conv(A): A ⊂ Orbit(y)

}, and

P∗(y) = {conv(A): A ⊂ Orbit(y) and |A| < ∞}

.

Since the norm ‖.‖ on H is generated by the dot product, for every P ∈ Fc(H) there exists an unique pointm(P ) ∈ P such that

∥∥m(P )

∥∥ = inf

{‖y‖: y ∈ P}.

Note that ‖m(P )‖ is exactly the distance between P and the origin of H, which we will denote by d(P ).In our next lemma, an ordered set A is said to be inversely well-ordered if every nonempty subset of A has a

maximal element.


Lemma 2.3. Let y ∈ Hfin+ ∪ H

fin− . Then, the set D∗(y) = {d(P ): P ∈ P∗(y)} is inversely well-ordered.

Proof. Take a point y ∈ Hfin+ , the case y ∈ H

fin− is completely analogous. Next, suppose, conversely to the conclu-sion of Lemma 2.3, that P∗(y) contains a sequence {Pi}∞i=1 such that {d(Pi)}∞i=1 is strictly increasing. According toProposition 2.2, Orbit(y) ⊂ {z ∈ H: ‖z‖ = ‖y‖}. Therefore, P ⊂ {z ∈ H: ‖z‖ � ‖y‖} for every P ∈ P∗(y). Thus, thesequence {d(Pi)}∞i=1 is bounded, and hence convergent. Put d = limi→∞ d(Pi).

Take a P ∈ P∗(y). According to Proposition 2.2, each ϕ, ϕ ∈ Bı(τ ), preserves the norm, hence it maps the pointm(P ) ∈ P to the point m(ϕ(P )) ∈ ϕ(P ). On the other hand, obviously, ϕ(Orbit(y)) = Orbit(y). Thus, by Proposi-tion 2.2, we have that

ϕ(P ) ∈ P∗(y),∥∥ϕ

(m(P )

)∥∥ = ∥∥m(P )∥∥ and ϕ

(m(P )

) = m(ϕ(P )

). (1)

Further, the coordinates of m(P ) are non-negative and only finitely many of them are positive, say at most k ofthem, k < ω, are positive. Hence, there exists a ϕ ∈ Bı(τ ) such that

ϕ(m(P )

)(α) � ϕ

(m(P )

)(α + 1) for α � k,

and ϕ(m(P ))(α) = 0 for α > k. That is, ϕ(m(P ))(α) � ϕ(m(P ))(α + 1) for every α < τ . Then, for every i ∈ N, thereexists a ϕi ∈ Bı(τ ) such that

ϕi

(m(Pi)

)(α) � ϕi

(m(Pi)

)(α + 1), α < τ.

Set Qi = ϕi(Pi), i ∈ N. According to (1), Qi ∈ P∗(y) and d(Qi) = d(Pi), i ∈ N. We are going to show that thesequence {m(Qi)}∞i=1 contains a convergent subsequence. To this end, let mi = m(Qi), i ∈ N, and let c = ∑

α<τ y(α).By definition, mi ∈ Qi = conv(Bi) for some finite Bi ⊂ Orbit(y), i.e. mi = ∑

b∈Biλb ·b where λb � 0 for every b ∈ Bi

and∑

b∈Biλb = 1. Observe that, for any b ∈ Bi ,

∑α<τ b(α) = ∑

α<τ y(α) = c. Therefore,∑

α<τ

mi(α) =∑

α<τ

∑

b∈Bi

λb · b(α) =∑

b∈Bi

λb

∑

α<τ

b(α) =∑

b∈Bi

λb · c = c.

Thus, according to Lemma 2.1, {mi}∞i=1 has a convergent subsequence. Without loss of generality, reducing ourattention to a subsequence of {Pi}∞i=1, we may assume that the sequence {mi}∞i=1 is itself convergent, and let m =limi→∞ mi . Note that ‖m‖ = d = limi→∞ d(Pi) and ‖mi‖ < ‖mi+1‖ < d holds for every i ∈ N.

In what follows, for every z ∈ H, with z �= 0, let γ (z) stand for the supporting hyperplain at the point z of theclosed ball {t ∈ H: ‖t‖ � ‖z‖}, and let γ+(z) and γ−(z) be the two open half-spaces of H determined by γ (z) so that0 ∈ γ−(z).

Let us observe that Bi ∩ γ−(m) �= ∅ for every i ∈ N. Indeed, if Qi ⊂ γ+(m) for some i ∈ N, then ‖mi‖ � ‖m‖ = d

which is impossible. So, for every i ∈ N there is a point bi ∈ Bi ∩ γ−(m). Now, on one hand, bi ∈ γ−(m) implies that〈bi − m,m〉 < 0. On another hand, Qi ⊂ γ+(mi) yields 〈bi − mi,mi〉 � 0. Therefore, for every i ∈ N, the followingholds

0 > 〈bi − m,m〉 = 〈bi − mi,mi〉 + 〈bi − mi − m,m − mi〉� 〈bi − mi − m,m − mi〉 = 〈bi,m − mi〉 − 〈m + mi,m − mi〉.

Since m = limi→∞ mi , the above implies that limi→∞〈bi − m,m〉 = 0. Indeed, we have ‖bi‖ = ‖y‖ because bi ∈Bi ⊂ Orbit(y), and therefore

∣∣〈bi,m − mi〉

∣∣ � ‖bi‖ · ‖m − mi‖ = ‖y‖ · ‖m − mi‖ −→

i→∞ 0.

Thus, we finally get that

〈bi,m〉 < ‖m‖2, i ∈ N, and limi→∞〈bi,m〉 = ‖m‖2. (2)

Let us show that (2) is impossible. To this end let supp(y) = {λ1, λ2, . . . , λn}, and put ti = y(λi), i = 1,2, . . . , n.Consider the set

M ={

n∑m(αi)ti : αi < τ and 1 � i � n

}

,

i=1


and then observe that {〈bi,m〉: i ∈ N} ⊂ M . To get this last contradiction, it now suffices to show that M is inverselywell-ordered, hence {〈bi,m〉: i ∈ N} must have a maximal element. This is what we will do till the end of this proof.So, for every k = 1,2, . . . , n, let

Mk ={

k∑

i=1

m(αi)ti : αi < τ and 1 � i � k

}

.

It is easily seen that {m(α): α < τ } is inversely well-ordered because m(α) � 0 for every α < τ , and∑

α<τ m2(α) <

∞. Then, obviously, M1 is inversely well-ordered as well. Suppose that, for some k < n, Mk is inversely well-orderedbut Mk+1 is not. Then, there exists a nonempty subset of Mk+1 without a maximal element, hence Mk+1 contains astrictly increasing sequence {μk+1

i }∞i=1. It is clear that μk+1i = μk

i + μi , where μki ∈ Mk and μi ∈ {m(α)tk+1: α < τ },

i ∈ N. Since Mk is inversely well-ordered, {μki }∞i=1 contains a decreasing subsequence {μk

ij}∞j=1, i.e. μk

ij� μk

ij+1for

j ∈ N. Therefore, for every j ∈ N,

μij = μk+1ij

− μkij

< μk+1ij+1

− μkij+1

= μij+1,

which contradicts the fact that {m(α)tk+1: α < τ } is inversely well-ordered. �Proposition 2.4. Let r, δ ∈ R and 0 < δ < r . Then, the diameter of any convex subset of H contained in Br+δ \ Br−δ

is less than or equal to 4√

rδ.

Proof. It follows by elementary geometric considerations. �Corollary 2.5. Let y ∈ H

fin+ ∪ Hfin− . Then, the set D(y) = {d(P ): P ∈P(y)} is inversely well-ordered.

Proof. It suffices to show that D∗(y) is dense in D(y). Indeed, assume it is so, but suppose that there exists asequence {di}∞i=1 ⊂ D(y), with di < di+1 for every i ∈ N. Then, for every i ∈ N, pick a d∗

i ∈ D∗(y) so closed to di

that d∗i < d∗

i+1. This gives rise to a strictly increasing sequence {d∗i }∞i=1 ⊂ D∗(y), which contradicts Lemma 2.3.

To show that D∗(y) is dense in D(y), take an arbitrary P ∈ P(y) and ε > 0. Let d = ‖m(P )‖, and let δ > 0 be suchthat the diameter of Pδ = Bd+δ ∩ P is less than ε (see, Proposition 2.4). Since P = conv(A) for some A ⊂ Orbit(y),there exists a finite set B ⊂ A with conv(B)∩Bd+δ �= ∅. Put Q = conv(B) ∈P∗(y). Clearly m(Q) ∈ Pδ , and therefore

∣∣d(Q) − d∣∣ = ∣∣∥∥m(Q)

∥∥ − ∥∥m(P )∥∥∣∣ �

∥∥m(Q) − m(P )∥∥ < ε. �

3. Proof of Theorem 1.3

Let X be a space, Y be a normed space, and let ε > 0. Recall that a map f :X → Y is an ε-selection for a mappingΦ :X → F(Y ) if Bε(f (x)) ∩ Φ(x) �= ∅ for every x ∈ X.

The following proposition summarizes a part of the technique developed in [4] for the proof of (a) ⇒ (b) ofTheorem 1.1.

Proposition 3.1. Let X be a space, and let Y be a Banach space. The following are equivalent:

(a) every l.s.c. mapping Φ :X → Fc(Y ), with Φ(x) bounded for each x ∈ X, has a continuous selection;(b) for every ε > 0, every l.s.c. mapping Φ :X → Fc(Y ), with Φ(x) bounded for each x ∈ X, has a continuous

ε-selection.

Proof. Suppose that Φ :X → Fc(Y ) is an l.s.c. mapping whose point-images are bounded. Then, (b) implies theexistence of a sequence {fn}∞n=1 of continuous maps fn :X → Y such that each fn, n ∈ N, is a 2−(n+1)-selectionfor Φ , and ‖fn(x) − fn+1(x)‖ < 2−n, x ∈ X. Namely, if fn is a continuous 2−(n+1)-selection for Φ , then considerthe mapping Φn :X → Fc(Y ), defined by Φn(x) = B2−(n+1) (fn(x)) ∩ Φ(x), x ∈ X. Obviously, each Φn(x), x ∈ X, isbounded, while, by [4, Propositions 2.3 and 2.5], Φn is l.s.c. Hence, by (b), Φn has a continuous 2−(n+2)-selectionfn+1 :X → Y . This fn+1 is a 2−(n+2)-selection for Φ such that ‖fn(x) − fn+1(x)‖ < 2−n, x ∈ X. Thus, we get a


Cauchy sequence {fn}∞n=1, which must converge to some continuous map f :X → Y because Y is a Banach space.The map f is a selection for Φ , which implies (a). Since (a) ⇒ (b) is obvious, the proof completes. �

The next proposition summarizes the idea of the verification of (b) in Proposition 3.1 in the case of Theorem 1.1,see [4, Lemma 4.1].

Proposition 3.2. Let X be a normal space, Y be a normed space, ε > 0, and let Φ :X → Fc(Y ) be an l.s.c. mappingsuch that the cover {Φ−1(Bε(y)): y ∈ Y } of X has a locally finite open refinement. Then, Φ has a continuous ε-se-lection.

Proof. Take a locally finite open refinement U of {Φ−1(Bε(y)): y ∈ Y }. According to [7, Lemma 1.2], there exists apartition of unity {gU : U ∈ U} on X index-subordinated to the cover U of X, i.e. g−1

U ((0,1]) ⊂ U , U ∈ U . For everyU ∈ U take a point yU ∈ Y , with U ⊂ Φ−1(Bε(yU )), and then define g :X → Y by

g(x) =∑{

gU(x) · yU : U ∈ U}, x ∈ X.

This g is a continuous ε-selection for Φ . �According to Propositions 3.1 and 3.2, to prove Theorem 1.3 it now suffices to prove the following theorem.

Theorem 3.3. Let X be a collectionwise normal space, Φ :X → Fc(H) be an l.s.c. mapping, with Φ(x) bounded foreach x ∈ X, and let ε > 0. Then, the cover {Φ−1(Bε(y)): y ∈ Y } of X has a locally finite open refinement.

This is, in fact, what we will do till the end of this section. To prepare for the proof of Theorem 3.3, we proceedwith a series of statements.

In what follows, for a family B of subsets of X and A ⊂ X, we use B � A to denote the trace of B on A, i.e.B � A = {B ∩ A: B ∈ B}.

Proposition 3.4. Let X be a collectionwise normal space, {Ai : i ∈ N} be a closed cover of X, and let B be an opencover of X such that, for every i ∈ N, B � Ai has a locally finite open refinement. Then B has a locally finite openrefinement as well.

Proof. By [7, Lemma 1.6] (see also [6]), for every i ∈ N, there is a locally finite open in X cover Vi of Ai such thatVi refines B. Whenever i ∈ N, take a functionally open set Oi ⊂ X, with Ai ⊂ Oi ⊂ ⋃

Vi , which is possible becauseX is normal. Then, {Oi : i ∈ N} is a countable functionally open cover of X and, by [5] (see, also, [2]), there exists alocally finite open cover {Ui :∈ N} of X, with Ui ⊂ Oi , i ∈ N. Finally,

⋃{Vi � Ui : i ∈ N} is a locally finite open coverof X which refines B.

For a mapping Φ :X → Fc(H) and ε > 0 we set BΦε = {Φ−1(Bε(y)): y ∈ H}. Also, to every A ⊂ X we assign the

possibly infinite number

ξΦ(A) = sup{d(Φ(x)

): x ∈ A

}.

Recall, that d(P ) = ‖m(P )‖ for P ∈Fc(H), where m(P ) is the unique point of P , with ‖m(P )‖ = inf{‖y‖: y ∈ P }.

Lemma 3.5. Let ε > 0, X be a collectionwise normal space and Φ :X → Fc(H) be an l.s.c. mapping with the propertythat

(∇) every nonempty closed set A ⊂ X contains another nonempty closed set δ(A) ⊂ A such that BΦε � δ(A) has a

locally finite open refinement, and ξΦ(A \ δ(A)) < ξΦ(A) provided δ(A) �= A.

Then, BΦε has a locally finite open refinement.


Proof. By induction, we are going to construct a σ -locally finite open refinement V = ⋃{Vγ : γ < β} of BΦε , for

some β < ω1, such that⋃

Vγ is functionally open for all γ < β . Then, the family {⋃Vγ : γ < β} will be a countablefunctionally open cover of X and, just like in the proof of Proposition 3.4, we may conclude that BΦ

ε has a locallyfinite open refinement.

To construct the cover V , let A0 = X. By hypothesis, there exists a nonempty closed δ(A0) ⊂ A0 such that BΦε �

δ(A0) has a locally finite open refinement. Then, by [7, Lemma 1.6], there exists a locally finite open in X cover V0of δ(A0) such that V0 refines BΦ

ε . Without loss of generality, we may assume that⋃

V0 is a functionally open subsetof X because δ(A0) ⊂ ⋃

V0 and X is normal.Suppose that, for some γ < ω1 and every λ < γ , a nonempty closed Aλ ⊂ X and a locally finite open family Vλ

are given such that Aλ = X \ ⋃{⋃Vμ: μ < λ}, ⋃Vλ ⊃ δ(Aλ),

⋃Vλ is functionally open and each element of Vλ

is contained in an element of BΦε . If

⋃{⋃Vλ: λ < γ } = X, then take β = γ and complete the induction. Otherwiseset Aγ = X \ ⋃{⋃Vλ: λ < γ } and, as before, get a locally finite open family Vγ such that

⋃Vγ ⊃ δ(Aγ ),

⋃Vγ is

functionally open and each element of Vγ is contained in an element of BΦε . Observe that, whenever γ1 < γ2 � γ ,

Aγ2 = X \⋃{⋃

Vλ: λ < γ2

}⊂ X \

⋃{⋃Vλ: λ � γ1

}

= Aγ1 \⋃

Vγ1 ⊂ Aγ1 \ δ(Aγ1)

and then, following (∇), ξΦ(Aγ2) < ξΦ(Aγ1). Hence, there must be some β < ω1, with⋃{⋃Vγ : γ < β} = X.

Otherwise we obtain a strictly decreasing transfinite sequence of real numbers {ξΦ(Aγ ): γ < ω1} which is impossible.This completes the proof. �Corollary 3.6. Let y ∈ H

fin+ ∪ Hfin− , ε > 0, X be a collectionwise normal space and Φ :X → P(y) be l.s.c. Then, Φ

satisfies condition (∇) of Lemma 3.5.

Proof. Take a nonempty closed subset A ⊂ X, and set

δ(A) = {x ∈ A: d

(Φ(x)

) = ξΦ(A)}

= {x ∈ A: Φ(x) ⊂ H \ BξΦ(A)

}.

Observe that δ(A) �= ∅ because, by Corollary 2.5, the set {d(Φ(x)): x ∈ A} has a maximal element being a nonemptysubset of D(y). By the same reason, if δ(A) �= A, then there is an a ∈ A \ δ(A), with d(Φ(a)) = ξΦ(A \ δ(A)) and,therefore, ξΦ(A \ δ(A)) < ξΦ(A). Moreover, δ(A) is closed because Φ is l.s.c. and δ(A) = A ∩ (X \ Φ−1(BξΦ(A))).Thus, it only remains to show that BΦ

ε � δ(A) has a locally finite open refinement. To this end set f (x) = m(Φ(x)),x ∈ δ(A), and take a locally finite open refinement U of {Bε(y): y ∈ H}. By [3, Theorem 4.1], f : δ(A) → H iscontinuous, hence {f −1(U): U ∈ U} is a locally finite open refinement of BΦ

ε � δ(A). �In our next lemma, for convenience, we will use Θ :Z � Y to denote that Θ is a mapping from Z to the nonempty

(not necessarily closed) subsets of Y . Also, for a subset B ⊂ Y of a metric space Y , we will use diam(B) to denotethe diameter of B with respect to the metric of Y .

Lemma 3.7. Let ε > 0, X be a collectionwise normal space and Ψ :X � H be an l.s.c. mapping with diam(Ψ (x)) <

ε/24 for all x ∈ X. Then BΨε admits a locally finite open refinement.

Proof. Put

Q = {y ∈ H

fin: y(α) ∈ Q for all α < τ and supp(y) ⊂ ω},

and

Xy = X \ Ψ −1(H \ Bε/24(Orbit(y))

)

= {x ∈ X: Ψ (x) ⊂ Bε/24(Orbit(y))

}, y ∈ Q.

Since Q is countable and⋃{Orbit(y): y ∈ Q} is dense in H, {Xy : y ∈ Q} is a closed countable cover of X. Then, by

Proposition 3.4, to prove our statement, it suffices to find a locally finite open refinement of BΨε � Xy for each y ∈ Q.


Turning to this purpose, fix an y ∈ Q and define Θy :Xy � Orbit(y) by the formula:

Θy(x) = {z ∈ Orbit(y): Bε/12(z) ∩ Ψ (x) �= ∅}

, x ∈ Xy.

Observe that, for each z ∈ Orbit(y),

Θ−1y

({z}) = {x ∈ Xy : z ∈ Θy(x)

}

= {x ∈ Xy : Bε/12(z) ∩ Ψ (x) �= ∅}

= Ψ −1(Bε/12(z)

) ∩ Xy.

Hence, Θ−1y ({z}) is relatively open in Xy , and therefore Θy is l.s.c. Moreover, the triangle inequality gives that

diam(Θy(x)) � diam(Bε/12(Ψ (x))) < 5ε/24.In what follows, to every z ∈ H we associate z+ ∈ H and z− ∈ H such that, for every α < τ ,

z+(α) ={

z(α) if z(α) � 0,

0 if z(α) � 0,

and

z−(α) ={

0 if z(α) � 0,

z(α) if z(α) � 0.

Then, for every x ∈ Xy , let

Θ+y (x) = {

z+: z ∈ Θy(x)}

and Θ−y (x) = {

z−: z ∈ Θy(x)}.

Observe that, for every x ∈ Xy , Θ+y (x) ⊂ Orbit(y+) and t ∈ Θ+

y (x) if and only if there is a z ∈ Θy(x), with z+ = t .Therefore

(Θ+

y

)−1({t}) = {x ∈ Xy : z+ = t for some z ∈ Θy(x)

}

=⋃{

Θ−1y (z): z ∈ Orbit(y) and z+ = t

}.

Thus (Θ+y )−1({t}) is open in Xy and hence Θ+

y :Xy � Orbit(y+) is l.s.c. Also, note that diam(Θ+y (x)) �

diam(Θy(x)) for all x ∈ Xy . Indeed, take t1, t2 ∈ Θ+y (x) and z1, z2 ∈ Θy(x), with z+

i = ti for i = 1,2. Then, ob-serve that

∣∣z+1 (α) − z+

2 (α)∣∣ �

∣∣z1(α) − z2(α)∣∣, for all α < τ,

and therefore

‖t1 − t2‖2 = ∥∥z+1 − z+

2

∥∥2 =∑

α<τ

(z+

1 (α) − z+2 (α)

)2

�∑

α<τ

(z1(α) − z2(α)

)2 = ‖z1 − z2‖2.

Analogously, Θ−y :Xy � Orbit(y−) is l.s.c. and diam(Θ−

y (x)) � diam(Θy(x)) for all x ∈ Xy . Now, define Θ+y :Xy →

P(y+) by setting Θ+y (x) = conv(Θ+

y (x)), x ∈ Xy , and Θ−y :Xy →P(y−) by setting Θ−

y (x) = conv(Θ−y (x)), x ∈ Xy .

Observe that, by [4, Propositions 2.3 and 2.6],

(1) Θ+y is l.s.c. because so is Θ+

y ,(2) diam(Θ+

y (x)) = diam(Θ+y (x)) � diam(Θy(x)) < 5ε/24 for all x ∈ Xy .

Exactly the same holds for Θ−y .

We finalize the proof in the following way. According to Proposition 3.2 and Corollary 3.6, both mappings Θ+y and

Θ−y have continuous ε/24-selections, respectively, f + :Xy → H and f − :Xy → H. Set f (x) = f +(x) + f −(x), for

every x ∈ Xy . Then, f :Xy → H is continuous and

f (x) ∈ ({f +(x)

} ∪ Θ+y (x) + {

f −(x)} ∪ Θ−

y (x)), x ∈ Xy.


Since the diameter of {f +(x)} ∪ Θ+y (x) (as well as the diameter of {f −(x)} ∪ Θ−

y (x)) is less than ε/24 +diam(Θ+

y (x)) < ε/24 + 5ε/24 = ε/4, it follows that the diameter of {f +(x)} ∪ Θ+y (x) + {f −(x)} ∪ Θ−

y (x) is lessthan ε/4 + ε/4 = ε/2. On the other hand, the intersection ({f +(x)} ∪ Θ+

y (x) + {f −(x)} ∪ Θ−y (x)) ∩ Bε/12(Ψ (x)) is

nonempty because it contains Θy(x), and therefore

f (x) ∈ Bε/2+ε/12(Ψ (x)

) = B7ε/12(Ψ (x)

) ⊂ B2ε/3(Ψ (x)

).

Thus, for every z ∈ H, f −1(Bε/3(z)) ⊂ Ψ −1(Bε(z)). Therefore, if U is a locally finite open refinement of {Bε/3(z):z ∈ H}, then {f −1(U): U ∈ U} is a locally finite open refinement of BΨ

ε � Xy . �Corollary 3.8. Let ε > 0, X be a collectionwise normal space and Φ :X → Fc(H) be a bounded l.s.c. mapping. ThenBΦ

ε has a locally finite open refinement.

Proof. By Lemma 3.5 it suffices to verify that Φ satisfies condition (∇) of that lemma. Take a nonempty closed setA ⊂ X. Since Φ is bounded, ξ = ξΦ(A) < ∞. Pick ρ > 0 such that 4

√ξρ < ε/24, and let δ(A) = A \ Φ−1(Bξ−ρ).

Obviously ξΦ(A\δ(A)) � ξ −ρ < ξ whenever δ(A) �= A. Next, define Ψ : δ(A) → H by letting Ψ (x) = Φ(x) ∩ Bξ+ρ

for every x ∈ δ(A). Note that Ψ is l.s.c. and diam(Ψ (x)) < ε/24 (see Proposition 2.4). Since δ(A) is collectionwisenormal being a closed subspace of X, Lemma 3.7 gives that BΨ

ε has a locally finite open refinement. To complete theproof observe that BΨ

ε refines BΦε � δ(A). �

Proof of Theorem 3.3. For every i ∈ N, let

Ai = X \ Φ−1(H \ Bi ) = {x ∈ X: Φ(x) ⊂ Bi

}.

Then, the family {Ai : i ∈ N} is a closed countable cover of X because Φ is l.s.c. and Φ(x) is bounded for eachx ∈ X. Let Φi = Φ�Ai , i ∈ N. By Corollary 3.8, each BΦi

ε = BΦε �Ai , i ∈ N, has a locally finite open refinement. Now

Lemma 3.4 completes the proof. �References

[1] M. Choban, S. Nedev, Continuous selections for mappings with generalized ordered domain, Math. Balkanica (N.S.) 11 (1–2) (1997) 87–95.[2] R. Engelking, General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989.[3] V. Gutev, S. Nedev, Continuous selections and reflexive Banach spaces, Proc. Amer. Math. Soc. 129 (6) (2001) 1853–1860.[4] E. Michael, Continuous selections: I, Ann. Math. 63 (1956) 562–590.[5] K. Morita, Star-finite coverings and star-finite property, Math. Japonicae 1 (1948) 60–68.[6] J. Nagata, Modern General Topology, second ed., North-Holland Math. Library, vol. 33, North-Holland, Amsterdam, 1985, p. 209.[7] S. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980) 291–317.[8] S. Nedev, A selection example, C. R. Acad. Bulgare Sci. 40 (11) (1987) 13–14.[9] I. Shishkov, Selections of l.s.c. mappings into Hilbert spaces, C. R. Acad. Bulgare Sci. 53 (7) (2000) 5–8.

[10] I. Shishkov, Σ -products and selections of set-valued mappings, Comment. Math. Univ. Carolinae 42 (1) (2001) 203–207.[11] I. Shishkov, Extensions of l.s.c. mappings into reflexive Banach spaces, Set-Valued Anal. 10 (1) (2002) 79–87.[12] I. Shishkov, Selections of set-valued mappings with hereditarily collectionwise normal domain, Topology Appl. 142 (1–3) (2004) 95–100.[13] K. Yosida, Functional Analysis, sixth ed., Springer-Verlag, Berlin, 1980, p. 126.



Existence and relaxation theorems for extreme continuous selectorsof multifunctions with decomposable values

A.A. Tolstonogov 1

Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences, P.O. Box 1233, Irkutsk 664033, Russia

Received 26 September 2006; received in revised form 31 March 2007; accepted 31 March 2007

Abstract

We present some extreme continuous selector theorems, synthesizing the author’s results; namely, we study existence and prop-erties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the spaceof Bochner integrable functions.© 2007 Elsevier B.V. All rights reserved.

MSC: 54C65; 54C60

Keywords: Extreme points; Strongly exposed points; Continuous selectors; Existence and relaxation theorems

1. Introduction

Continuous selectors theory is a classical mathematical field which has various and numerous applications. Thefundamental results in this area were obtained in the mid of 1950’s by E. Michael. An important class of multifunctionsconsists of maps with closed, not necessarily convex, decomposable values. The study of existence of continuousselectors for such multifunctions was initiated by Antosiewicz and Cellina’s ideas [1] and continued in [9,7,10,2,13].The main purpose of this paper is to unify selector theorems for multifunctions whose values are extreme pointsof closed convex decomposable sets in the space of Bochner integrable functions. As is well known, in general thecollection of extreme points of a closed convex set is not closed. This means that in the study of our problems wecannot exploit convergence arguments used in [1,9,7]. Therefore we develop a different method, based on Baire’scategory theorem. It should be mentioned that this technique has been used in [6] in order to prove the existence ofso-called directionally continuous selectors for a special class of multifunctions with nonclosed nonconvex values.

For the first time the existence of extreme continuous selectors and relaxation theorems were proved by the Bairecategory method for some class of multifunctions in [15,16], while generalizations are contained in [17,24,25]. In thispaper we propose some further extension of these results.

E-mail address: [email protected] Supported in part RFBR grant 06-01-00247-a.


A.A. Tolstonogov / Topology and its Applications 155 (2008) 898–905 899

2. Notations and definitions

Let (X,‖·‖) be a separable Banach space, (X′,‖·‖) be its topologically dual space, and 〈·, ·〉 be the canonicalduality between X and X′. We use the notations w − X and w − X′ for the spaces X and X′ furnished with theweak topologies σ(X,X′) and σ(X′,X), respectively (see [3]). Also, let M and T be locally compact separablemetric spaces such that T is equipped, moreover, with a nonnegative finite nonatomic Radon measure μ0 and theσ -algebra Σ of μ0-measurable subsets. In what follows R is the number line, and χ(A) is the characteristic functionof a set A.

By Lp(T ,X), 1 � p < ∞, we mean the Banach space of equivalence classes of Bochner integrable functionsfrom T to X with the standard norm ‖·‖p .

A set K ⊂ Lp(T ,X) is said to be decomposable if χ(E)u + χ(T \E)v ∈ K , whenever u,v ∈ K and E ∈ Σ .Recall that the closure of a decomposable set is also decomposable. Denote by dLp(T ,X) the family of all non-

empty decomposable closed bounded subsets of Lp(T ,X) and by cdLp(T ,X) the family of all nonempty convexdecomposable closed bounded subsets of Lp(T ,X). We consider the space dLp(T ,X) supplied with the Hausdorffmetric generated by the topology of the space Lp(T ,X).

For a subset K of a Banach space Y the symbol coK means its convex hull and coK means its closed convex hull.By extK we denote the family of all extreme points of a convex closed bounded set K .

For K ⊂ X, x′ ∈ X′, x′ = 0 and α > 0 let us set C(K,x′) := sup{〈x, x′〉;x ∈ K} and C(K,x′, α) := {x ∈ K;〈x, x′〉 > C(K,x′) − α}.

Definition 2.1. (See [5].) Let K ⊂ X be a convex closed bounded subset and x ∈ K . The point x is said to bestrongly exposed point of K if there exists x′ ∈ X′ such that 〈x, x′〉 > 〈y, x′〉 whenever y = x, y ∈ K , y = x, and{C(K,x′, α);α > 0} is a neighborhood base of x in K with respect to the norm topology.

Denote by stK the set of all strongly exposed points of K .

Definition 2.2. (See [5].) Let (Ω,F ,μ) be a probability space. For an X-valued measure m on (Ω,F) absolutelycontinuous with respect to μ its average rang is defined by

AR(m) ={

m(A)

μ(A);A ∈F ,μ(A) > 0

}.

Definition 2.3. (See [5].) A closed bounded convex set K ⊂ X is said to have the Radon–Nikodym Property for(Ω,F ,μ) if for each X-valued measure m on F absolutely continuous with respect to μ and with average rangeAR(m) contained in K there exists a function f ∈ L1(Ω,X) such that m(A) = ∫

Af dμ for each A ∈ F . The set K is

said to have the Radon–Nikodym Property (briefly RNP) if K has the Radon–Nikodym Property for each probabilityspace (Ω,F ,μ).

By using Theorem 2.3.6 and Corollary 3.5.7 in [5] we obtain

Proposition 2.4. Let K be a closed convex bounded set in X. Then the following properties are equivalent:

(1) K has the RNP;(2) K = co(stK).

It is well known [5], that every convex weakly compact set in X has the RNP. If K ⊂ X is closed bounded, andcoK has the RNP then st(coK) ⊂ K .

By RNcdLp(T ,X),1 � p < ∞, we denote the family of all sets from cdLp(T ,X) with RNP.Let P(y) be a continuous seminorm in Lp(T ,X) and

∂P = {y′ ∈ L′

p(T ,X); 〈y, y′〉 � P(y),∀y ∈ Lp(T ,X)}

be its subdifferential.

900 A.A. Tolstonogov / Topology and its Applications 155 (2008) 898–905

Definition 2.5. (See [23].) A continuous seminorm P(y) on Lp(T ,X), 1 � p < ∞, is said to have the scalar com-pactness property (briefly SCP) if for each x ∈ X the set of numerical functions

{z ∈ L1(T ,R); z(t) = 〈x, y′(t)〉, y′ ∈ ∂P

}

is relatively compact in L1(T ,R).

Definition 2.6. (See [18].) A continuous seminorm P(y) on Lp(T ,X), 1 � p < ∞, is said to have the relaxationproperty (briefly RP) on the space RNcdLp(T ,X) if for each K ∈ RNcdLp(T ,X),u ∈ K and ε > 0 there existsv ∈ stK such that P(u − v) < ε.

Examples of seminorms with SCP and RP are considered in Section 5.We assume that all multifunctions under consideration have nonempty values.A multifunction F :T → X is called measurable [12] if for each closed subset U ⊂ X the set {t ∈ T ;F(t)∩U = ∅}

is measurable.A multifunction F from a topological space Y into a topological space Z is called lower semicontinuous at a

point y0 ∈ Y if for each open set U ⊂ Z such that F(y0) ∩ U = ∅ there exists a neighborhood V (y0) of y0 such thatF(y) ∩ U = ∅ for all y ∈ V (y0).

Let F :T → X be a measurable multifunction. For 1 � p < ∞ we define the set

Sp(F ) = {f ∈ Lp(T ,X);f (t) ∈ F(t) a.e.

}.

It is well known [11] that for Γ ∈ dLp(T ,X) there exists a unique (up to a set of measure zero) measurablemultifunction F(Γ ) :T → X with closed bounded values such that

Γ = Sp(F(Γ )

). (2.1)

We refer to the map F(Γ ) :T → X satisfying (2.1) as to the set-valued representation of the set Γ .

3. Preliminaries

We recall some results being used in the next section.Let {x′

s}∞1 be a countable σ(X′,X)-dense balanced subset of {x′ ∈ X′; ‖x′‖ � 1}. For each Γ ∈ cdLp(T ,X), u ∈ Γ

and x′s define the function

Ds(Γ,u) := sup{〈y − z, x′

s · χ(T )〉;y, z ∈ Γ,u = (y + z)/2},

where⟨y − z, x′

s · χ(T )⟩ :=

∫

T

⟨y(t) − z(t), x′

s · χ(T )(t)⟩dμ0.

Proposition 3.1. (See [24].) Let Γ ∈ cdLp(T ,X) and u ∈ Γ . Then the following statements hold:

(i) u ∈ extΓ if and only if Ds(Γ,u) = 0 for every s � 1;(ii) if Γn,Γ ∈ cdLp(T ,X), n � 1, Γn converges to Γ with respect to the Hausdorff metric, and un ∈ Γ , n � 1,

converges to u in Lp(T ,X) then

limn→∞ supDs(Γn,un) � Ds(Γ,u), s � 1.

Proposition 3.2. (See [21].) Let Γ ∈ cdLp(T ,X), 1 � p < ∞ and F(Γ ) be its set-valued representation. A pointu ∈ Γ is strongly exposed if and only if u(t) is a strongly exposed point of the set F(Γ )(t) for almost all t ∈ T .

It follows from Proposition 3.2 that for Γ ∈ cdLp(T ,X) the set stΓ is decomposable.For the sake of briefness, a continuous with respect to the topology of the space Lp(T ,X) function u :M →

Lp(T ,X) will be called Lp-continuous.


Proposition 3.3. Let Γ :M → RNcdLp(T ,X) be a multifunction which is continuous with respect to the Hausdorffmetric. Then the multifunction ξ → stΓ (ξ) is lower semicontinuous and for each ξ0 ∈ M , u0 ∈ stΓ (ξ) there existsan Lp-continuous selector u(ξ) of stΓ (ξ) satisfying u(ξ0) = u0, where stΓ (ξ) is the closure of the set stΓ (ξ).

Proof. From Lemma 2 [14] it follows that the multifunction stΓ (ξ) is lower semicontinuous. By using Proposi-tion 3.2 we obtain that the multifunction stΓ (ξ) is lower semicontinuous with closed decomposable values. Now thestatement follows from [7,16]. �Theorem 3.4. Let Γ :M → RNcbLp(T ,X) be a continuous multifunction (with respect to the Hausdorff metric);the functions Φi :M → (0,+∞), i = 1,2, be lower semicontinuous, and u(ξ) be an Lp-continuous selector of themultifunction stΓ (ξ). Then for any s � 1 there exists an Lp-continuous selector v(ξ) of the multifunction stΓ (ξ)

such that∥∥u(ξ) − v(ξ)

∥∥p

< Φ1(ξ), ξ ∈ M,

and

Ds(Γ (ξ), v(ξ)

)< Φ2(ξ), ξ ∈ M.

This theorem follows from Theorem 6.6 [24] because a decomposable set coincides with its decomposablehull [24].

4. Main results

Let CC(M,Lp) be the space of all continuous functions from M to Lp(T ,X) endowed with the topology ofuniform convergence on compact subsets of M . Then CC(M,Lp) is the complete metrizable space. Let us take acontinuous (with respect to the Hausdorff metric) multifunction Γ :M → RNcdLp(T ,X). Denote by CC(stΓ ) thefamily of all Lp-continuous selectors of the multifunction stΓ (ξ) endowed with the topology of uniform convergenceon compact subsets of M . It follows from Proposition 3.3 that CC(stΓ ) is not empty. Since CC(stΓ ) is closed inCC(M,Lp), it is the complete metric space.

Existence Theorem. Let Γ :M → RNcdLp(T ,X) be a continuous multifunction (with respect to the Hausdorffmetric). Then for each Lp-continuous selector u(ξ) of stΓ (ξ) and each lower semicontinuous function Φ :M →(0,+∞) there exists an Lp-continuous selector v(ξ) of the multifunction extΓ (ξ) such that

∥∥u(ξ) − v(ξ)∥∥

p< Φ(ξ), ξ ∈ M,

and

v(ξ) ∈ extΓ (ξ) ∩ stΓ (ξ), ξ ∈ M.

Let us divide the proof of this theorem into several steps.For an Lp-continuous selector u(ξ) of the multifunction stΓ (ξ) and for a function Φ :M → (0,+∞) let us denote

by HU the closure in CC(stΓ ) of the set of all Lp-continuous selectors v(ξ) of stΓ (ξ), satisfying the inequality

∥∥u(ξ) − v(ξ)∥∥

p<

Φ(ξ)

2, ξ ∈ M.

Since HU is not empty and closed in CC(stΓ ), it is also complete.Fix η > 0, s � 1 and consider the set

Hsη = {

v ∈HU ;Ds(Γ (ξ), v(ξ)

)< η, ξ ∈ M

}.

Lemma 4.1. The set Hsη is a Gδ subset of HU .


Proof. According to Theorem 3.4 Hsη is not empty. Since M is locally compact separable metric space, there exists

an increasing (by inclusion) sequence of compact sets Kn ⊂ M , n � 1, such that M = ⋃∞n=1 Kn, and each compact

set Q ⊂ M is contained in Kn for some n � 1 [4]. Let us consider the set

Hsη(n) := {

v ∈HU ;Ds(Γ (ξ), v(ξ)

)< η, ξ ∈ Kn

}, n � 1.

Then

Hsη =

∞⋂

n=1

Hsη(n).

The lemma will be proved if we show that Hsη(n), n � 1, are open subsets of HU .

Fix n � 1. It is enough to prove that the set HU\Hsη(n) is closed. Let uk ∈ HU\Hs

η(n), k � 1, be an arbitrarysequence converging to v ∈HU with respect to the topology of the space CC(stΓ ). Then for each k � 1 there exists apoint ξk ∈ Kn such that Ds(Γ (ξk), uk(ξk)) � η. By compactness of Kn, passing to a subsequence if necessary, we canassume that the sequence ξk , k � 1, converges to a point ξ0 ∈ Kn. Since uk , k � 1, converges to v uniformly on Kn,the sequence uk(ξk), k � 1, converges to v(ξ0). According to Proposition 3.1 we obtain that Ds(Γ (ξ0), v(ξ0)) � η.Therefore, v ∈HU\Hs

η(n) and the set Hsη(n) is open in HU .

Lemma 4.2. The set Hsη is a dense subset of HU .

Proof. Let Q ⊂ M be an arbitrary compact set, w ∈HU and ε > 0. We have that Q ⊂ Kn for some n � 1. Accordingto the definition of HU there exists v1 ∈ CC(stΓ ) such that

∥∥u(ξ) − v1(ξ)∥∥

p< Φ(ξ)/2, ξ ∈ M, and

∥∥v1(ξ) − w(ξ)∥∥

p< ε, ξ ∈ Kn.

Set

d(ξ) ={

min{Φ(ξ)/2 − ‖u(ξ) − v1(ξ)‖p, ε − ‖v1(ξ) − w(ξ)‖p}, ξ ∈ Kn,

Φ(ξ)/2 − ‖u(ξ) − v1(ξ)‖p, ξ ∈ M\Kn.

The function d(ξ) is lower semicontinuous and d(ξ) > 0 for all ξ ∈ M . By Lemma 3.6 [8] there exists a continuousfunction c(ξ), 0 < c(ξ) < d(ξ), ξ ∈ M . It follows from Theorem 3.4 that there exists an Lp-continuous selector v(ξ)

of the multifunction stΓ (ξ) such that∥∥v1(ξ) − v(ξ)

∥∥p

< c(ξ) and Ds(Γ (ξ), v(ξ)

)< η for all ξ ∈ M.

Then∥∥u(ξ) − v(ξ)

∥∥p

�∥∥u(ξ) − v1(ξ)

∥∥p

+ ∥∥v1(ξ) − v(ξ)∥∥

p<

∥∥u(ξ) − v1(ξ)∥∥

p+ c(ξ) < Φ(ξ)/2, ξ ∈ M.

Analogously, one can obtain that∥∥w(ξ) − v(ξ)

∥∥p

< ε, ξ ∈ Kn.

Therefore, v ∈Hsη and Hs

η is dense in HU . �Lemma 4.3. The set

⋂n�1

⋂s�1 Hs

1/n is dense in HU .

This result follows from Lemmas 4.1, 4.2 and from Baire’s category theorem [4].

Proof of Existence Theorem. Let v ∈ ⋂n�1

⋂s�1 Hs

1/n. Then Ds(Γ (ξ), v(ξ)) < 1/n for all ξ ∈ M , s � 1. There-fore Ds(Γ (ξ), v(ξ)) = 0, ξ ∈ M , s � 1, and, it follows from Proposition 3.1 that v(ξ) ∈ extΓ (ξ), ξ ∈ M . Sincev ∈HU , we have

∥∥u(ξ) − v(ξ)∥∥

p� Φ(ξ)/2 < Φ(ξ), ξ ∈ M,

and v(ξ) ∈ stΓ (ξ), ξ ∈ M . This concludes the proof. �


Corollary 4.4. Let F :M → dLp(T ,X) be such that the multifunction coF(ξ), ξ ∈ M , satisfies all the assumptions ofthe Existence Theorem. Then for each Lp-continuous selector u(ξ) of st coF(ξ) and for each lower semicontinuousfunction Φ :M → (0,+∞) there exists an Lp-continuous selector v(ξ) of the multifunction st coF(ξ) such that

∥∥u(ξ) − v(ξ)

∥∥

p< Φ(ξ), ξ ∈ M,

and

v(ξ) ∈ F(ξ) ∩ ext coF(ξ), ξ ∈ M.

This corollary follows from the Existence Theorem because st coF(ξ) ⊂ F(ξ) for all ξ ∈ M .

Relaxation Theorem. Let Γ :M → RNcbLp(T ,X) be a continuous multifunction (with respect to the Hausdorffmetric), and P :Lp(T ,X) → R be a continuous seminorm with SCP and RP on the space RNcdLp(T ,X). Then foreach Lp-continuous selector u(ξ) of Γ (ξ) and for each lower semicontinuous function Φ :M → (0,+∞) there existsan Lp-continuous selector v(ξ) of the multifunction stΓ (ξ) such that

v(ξ) ∈ extΓ (ξ), ξ ∈ M,

and

P(u(ξ) − v(ξ)

)< Φ(ξ), ξ ∈ M.

Proof. Since P(x) is a continuous seminorm on Lp(T ,X), there exists c > 0 with

P(x) � c‖x‖p, x ∈ Lp(T ,X).

Set

Bp := {x ∈ Lp(T ,X);P(x) < 1

}.

It follows from the relaxation property of P(x) that

G(ξ) = stΓ (ξ) ∩(

u(ξ) + Φ(ξ)

2· Bp

)= ∅, ξ ∈ M.

Since the multifunction stΓ (ξ) is lower semicontinuous with closed decomposable values and P(x) has SCP, accord-ing to Theorem 3.1 [10], there exists an Lp-continuous selector v1(ξ) of stΓ (ξ) such that

P(u(ξ) − v1(ξ)

)< Φ(ξ)/2, ξ ∈ M. (4.1)

By using of the Existence Theorem, we find an Lp-continuous selector v(ξ) of stΓ (ξ) such that∥∥v(ξ) − v1(ξ)

∥∥p

< Φ(ξ)/2c, ξ ∈ M, (4.2)

and

v(ξ) ∈ extΓ (ξ), ξ ∈ M. (4.3)

Now the statements of theorem follow from (4.1)–(4.3). �Corollary 4.5. Assume that a multifunction F :M → dLp(T ,X) and a continuous seminorm P(x) satisfy all the as-sumptions of the Relaxation Theorem. Then for each Lp-continuous selector u(ξ) of coF(ξ) and for each lower semi-continuous function Φ :M → (0,+∞) there exists an Lp-continuous selector v(ξ) of the multifunction st coF(ξ)

such that

v(ξ) ∈ F(ξ) ∩ ext coF(ξ), ξ ∈ M,

and

P(u(ξ) − v(ξ)

)< Φ(ξ), ξ ∈ M.

This result follows from the Relaxation Theorem and from the inclusion st coF(ξ) ⊂ F(ξ), ξ ∈ M .


5. Examples

Let Ω ⊂ Rn be a bounded domain with the Lebesgue measure μ0. Denote by D the family of sets E ⊂ Rn of theform E = [a1, b1] × · · · × [an, bn].

Let Q ⊂ L′p(Ω,X), 1 � p < ∞, be relatively compact set.

Proposition 5.1. (See [25].) A function

P(x) = supx′∈QE∈D

∣∣∣∣

∫

E∩Ω

⟨x(t), x′(t)

⟩dμ0

∣∣∣∣, x ∈ Lp(Ω,X)

is the continuous seminorm in Lp(Ω,X) with SCP and RP on the space RNcdLp(Ω,X).

Corollary 5.2. If x′i ∈ L′

p(Ω,X), 1 � p < ∞, i = 1, . . . ,m, then the function

P(x) = sup1�i�m

∣∣∣∣

∫

Ω

⟨x(t), x′

i (t)⟩dμ0

∣∣∣∣, x ∈ Lp(Ω,X)

is the continuous seminorm in Lp(Ω,X) with SCP and RP on the space RNcdLp(Ω,X).

Proposition 5.3. (See [25].) The function

P(x) = supE∈D

∥∥∥∥

∫

E∩Ω

x(t)dμ0

∥∥∥∥, x ∈ Lp(Ω,X), 1 � p < ∞, (5.1)

is the continuous norm in Lp(Ω,X) with SCP and RP on the space RNcdLp(Ω,X).

Corollary 5.4. If T = [0,1] ⊂ R, then the function

P(x) = sup0�t�t ′�1

∥∥∥∥∥

t ′∫

t

x(t)dμ0

∥∥∥∥∥, x ∈ Lp(T ,X), 1 � p < ∞, (5.2)

is the continuous norm in Lp(T ,X) with SCP and RP on the space RNcdLp(T ,X).

Let T = [0,1] and S(t, s), (t, s) ∈ Δ, Δ := {(t, s) ∈ T × T ;0 � s � t � 1} be a strongly continuous semigroup ofcontinuous linear operators from X into X (a C0-semigroup).

Proposition 5.5. (See [18].) The function

P(x) = sup0�t�1

∥∥∥∥∥

t∫

0

S(t, s)x(s)ds

∥∥∥∥∥, x ∈ Lp(T ,X), 1 � p < ∞, (5.3)

is the continuous norm in Lp(T ,X) with SCP and RP on the space RNcdLp(Ω,X).

Corollaries 4.4, 4.5 and the norms (5.1)–(5.3) are crucial in some sense in the theory of differential inclusions withnonconvex right-hand side [15,16,19,25] and in nonconvex optimal control problems [20,22].

References

[1] H.A. Antosiewicz, A. Cellina, Continuous selectors and differential relations, J. Differential Equations 19 (1975) 386–398.[2] S.M. Ageev, D. Repovs, On selections theorems with decomposable values, Topol. Methods Nonlinear Anal. 15 (2000) 385–399.[3] N. Bourbaki, Topological Vector Spaces, Izdat. Inostr. Lit., Moscow, 1959 (Russian translation).[4] N. Bourbaki, General Topology. Functional Spaces, Izdat. Nauka, Moscow, 1975 (Russian translation).[5] R.D. Bourgin, Geometric Aspects of Convex sets with the Radon–Nikodym Property, Lecture Notes in Math., vol. 993, Springer-Verlag, 1983.


[6] A. Bressan, Differential inclusions with non-closed, non-convex right-hand side, Differential Integral Equations 3 (1990) 633–638.[7] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988) 69–86.[8] F.S. De Blasi, G. Pianigiani, Remarks on Hausdorff continuous multifunctions and selections, Comment. Math. Univ. Carolinae 24 (1983)

553–561.[9] A. Fryszkowski, Continuous selections for a class of nonconvex multivalued maps, Studia Math. 76 (1983) 163–174.

[10] V.V. Goncharov, A.A. Tolstonogov, Continuous selections for a family of nonconvex-valued mappings with noncompact domain, SiberianMath. J. 35 (1994) 479–494.

[11] F. Hiai, H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182.[12] C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975) 53–72.[13] H.T. Nguyen, M. Juniewicz, J. Zieminska, CM-selectors for a pairs of oppositely semicontinuous multivalued maps with Lp-decomposable

values, Studia Math. 144 (2001) 135–151.[14] S.I. Suslov, Nonlinear “bang–bang” principle in a Banach space, Siberian Math. J. 33 (1992) 675–685.[15] A.A. Tolstonogov, Extreme selectors of multivalued maps and the “bang–bang” principle for evolution inclusions, Soviet. Math. Dokl. 43

(1991) 481–485.[16] A.A. Tolstonogov, Extreme continuous selectors of multivalued maps and their applications, J. Differential Equations 122 (1995) 161–180.[17] A.A. Tolstonogov, Continuous selections of multivalued maps with non-convex, non-closed decomposable values, Sb. Math. 187 (1996)

745–766.[18] A.A. Tolstonogov, Lp-continuous selections of fixed points of multifunctions with decomposable values. II: Relaxation theorems, Siberian

Math. J. 40 (1999) 991–1003.[19] A.A. Tolstonogov, Lp-continuous selections of fixed points of multifunctions with decomposable values. III: Applications, Siberian Math.

J. 40 (1999) 1173–1187.[20] A.A. Tolstonogov, Relaxation in non-convex optimal control problems described by first-order evolution equations, Sb. Math. 190 (1999)

1689–1714.[21] A.A. Tolstonogov, Strongly exposed points of decomposable sets in spaces of Bochner integrable functions, Math. Notes 71 (2002) 267–275.[22] A.A. Tolstonogov, Bogolyubov’s theorem under constraints generated by a lower semicontinuous differential inclusion, Sb. Math. 196 (2005)

263–285.[23] A.A. Tolstonogov, V.V. Goncharov, On sublinear functionals defined on the space of Bochner integrable functions, Siberian Math. J. 35 (1994)

178–188.[24] A.A. Tolstonogov, D.A. Tolstonogov, Lp-continuous extreme selectors of multifunctions with decomposable values: existence theorems,

Set-Valued Anal. 4 (1996) 173–203.[25] A.A. Tolstonogov, D.A. Tolstonogov, Lp-continuous extreme selectors of multifunctions with decomposable values: relaxation theorems,

Set-Valued Anal. 4 (1996) 237–269.



Parametric bing and Krasinkiewicz maps

Vesko Valov 1

Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7, Canada

Received 6 November 2006; received in revised form 20 November 2006; accepted 20 November 2006

Abstract

It is shown that if f :X → Y is a perfect map between metrizable spaces and Y is a C-space, then the function space C(X, I)

with the source limitation topology contains a dense Gδ-subset of maps g such that every restriction map gy = g|f −1(y), y ∈ Y ,satisfies the following condition: all fibers of gy are hereditarily indecomposable and any continuum in f −1(y) either contains acomponent of a fiber of gy or is contained in a fiber of gy .Crown Copyright © 2007 Published by Elsevier B.V. All rights reserved.

MSC: primary 54C65; secondary 54F15

Keywords: Hereditarily indecomposable space; Selections for set-valued maps; C-spaces

1. Introduction

All spaces in the paper are assumed to be metrizable and all maps continuous. Unless stated otherwise, any functionspace C(X,M) is endowed with the source limitation topology whose neighborhood base at a given function f ∈C(X,M) consists of the sets

B�(f, ε) = {g ∈ C(X,M): �(g,f ) < ε

},

where � is a fixed compatible metric on M and ε :X → (0,1] runs over continuous functions into (0,1]. The symbol�(f,g) < ε means that �(f (x), g(x)) < ε(x) for all x ∈ X. The source limitation topology does not depend on themetric � [15] and has the Baire property provided M is completely metrizable [20]. Obviously, this topology coincideswith the uniform convergence topology when X is compact.

A compactum is called a Bing space if each of its subcontinua is hereditarily indecomposable. A map g is said to bea Bing map [16] if all fibers of g are Bing spaces. Following Krasinkiewicz [13], we say that space M is a free space iffor any compactum X the function space C(X,M) contains a dense subset consisting of Bing maps. The class of freespaces is quite large, it contains all n-manifolds [13], in particular, the unit interval [16], all locally finite polyhedrons[24], as well as all manifolds modeled on Menger cubes Mn

2n+1 or Nöbeling spaces Nn2n+1 [24] (n � 1). This class

also contains 1-dimensional locally connected continua [24]. Surjective Bing maps were considered in [12].In the present paper we provide a parametric versions of the above results concerning free spaces.

E-mail address: [email protected] The author was partially supported by NSERC Grant 261914-03.

0166-8641/$ – see front matter Crown Copyright © 2007 Published by Elsevier B.V. All rights reserved.doi:10.1016/j.topol.2006.11.011

V. Valov / Topology and its Applications 155 (2008) 906–915 907

Theorem 1.1. Let f :X → Y be a perfect map with Y being a strongly countable-dimensional space. Then, for everycomplete ANR free space M , the function space C(X,M) contains a dense Gδ-set of maps g such that all restrictionsg|f −1(y), y ∈ Y , are Bing maps. Moreover, if in addition M is a closed convex subset of a Banach space, then thesame conclusion remains true provided Y is a C-space.

Recall that X is a C-space if for any sequence {νn}∞n=1 of open covers of X there exists a sequence {γn}∞n=1 ofdisjoint open families in X such that each γn refines νn and

⋃∞n=1 γn is a cover of X. Every strongly countable-di-

mensional space (i.e. a space which is a union of countably many closed finite-dimensional subsets), as well as everycountable-dimensional space (a countable union of 0-dimensional subsets) is a C-space [7] and there exists a compactC-space which is not countable-dimensional [22].

We also consider the so-called Krasinkiewicz maps [17]. A map g :X → M between compacta is called aKrasinkiewicz map if every continuum in X is either contained in a fiber of g or contains a component of a fiberof g. Krasinkiewicz [14] proved that, for any compactum X, the Krasinkiewicz maps from space C(X, I) form adense subset. Levin and Lewis [17] established that this set is also a Gδ-subset of C(X, I). Our second theorem in thepresent paper provides a parametric version of the Levin–Lewis result.

Theorem 1.2. Let f :X → Y be a perfect map with Y being a C-space. Then C(X, I) contains a dense Gδ-subset ofmaps g such that all restrictions g|f −1(y), y ∈ Y , are Bing and Krasinkiewicz maps.

The proof of Theorem 1.1 is given in Section 2. It is based on few selection theorems: the convex-valued selectiontheorem of Michael [19], a selection theorem for finite-dimensional spaces due to Gutev [9, Theorem 3.1], and aselection theorem for C-spaces established by Uspenskij [25, Theorem 1.3]. Section 3 is devoted to the proof ofTheorem 1.2. In the final Section 4 we discuss some possible generalizations of the Uspenskij selection theorem forC-spaces which would imply the validity of Theorem 1.1 with Y being a complete ANR free space having the C-space property. In this section we also present a version of Theorem 1.1, where M is a free completely metrizableLCn-space. As an application of Theorem 1.1, we establish a result concerning the Bula property of perfect open mapsbetween metrizable spaces.

2. Proof of Theorem 1.1

Following [16], we consider the family D of all 4-tuples (F0,F1,V0,V1) such that F0 and F1 are disjoint closedsubsets of the Hilbert cube Q and V0,V1 disjoint neighborhoods of F0 and F1, respectively, in Q. A set P ⊂ Q isD-crooked (see [3] and [16]) for D = ((F0,F1,V0,V1) ∈ D if there exists a neighborhood G of P in Q such that forevery map h : I = [0,1] → G with h(0) ∈ F0 and h(1) ∈ F1 there are t0, t1 ∈ I such that 0 < t0 < t1 < 1, h(t0) ∈ V1

and h(t1) ∈ V0. According to [3] and [16], D has the following properties:

• a compactum P ⊂ Q is a Bing space if and only if P is D-crooked for every D ∈ D;• there is a sequence {Di}i�1 ⊂ D such that for every compactum P ⊂ Q, P is a Bing space iff P is Di -crooked

for every i.

A map g :P ⊂ Q → M is called D-crooked, where D ∈ D, if all fibers of g are D-crooked. We fix a sequence{Di}i�1 ⊂ D with the above property and a map h :X → Q such that h embeds every fiber f −1(y), y ∈ Y (sucha map exists by [21, Proposition 9.1]). For given y ∈ Y we denote by hy the restriction map h|f −1(y) :f −1(y) →h(f −1(y)). Let Hi (y), where y ∈ Y and i � 1, be the set of all g ∈ C(X,M) such that the map g ◦h−1

y :h(f −1(y)) →M is Di -crooked. If F ⊂ Y , then Hi (F ) is the intersection of all Hi (y), y ∈ F . Obviously, if Y = ⋃∞

m=1 Ym with eachYm being closed in Y , the set H = ⋂∞

i,m=1 Hi (Ym) consists of maps g ∈ C(X,M) such that all restriction mapsg|f −1(y), y ∈ Y , are Bing maps. Therefore, it suffices to show that Hi (F ) is open and dense in C(X,M) with respectto the source limitation topology for any closed F ⊂ Y in any of the following cases: (i) F is finite-dimensionalprovided Y is strongly countable-dimensional and M is a complete free ANR-space; (ii) F is arbitrary if Y is a C-space and M a closed convex free subspace of a Banach space.

908 V. Valov / Topology and its Applications 155 (2008) 906–915

2.1. Every Hi (F ) is open in C(X,M)

In this subsection we suppose that M is a fix completely metrizable ANR and Y a metrizable space.

Lemma 2.1. The space M admits a complete bounded metric � generating its topology and satisfying the followingcondition: If Z is a paracompact space, A ⊂ Z a closed set and ϕ :Z → M a map, then for every function α :Z →(0,1] and every map g :A → M with �(g(z),ϕ(z)) < α(z)/8 for all z ∈ A, there exists a map g :Z → M extending g

such that �(g(z), ϕ(z)) < α(z) for all z ∈ Z.

Proof. We embed M as a closed subset of a Banach space E. Since M ∈ ANR, there exist a neighborhood W ofM in E and a retraction r :W → M . For every open U ⊂ M let T (U) = W\r−1(M\U). Obviously, T (U) ⊂ W isopen, T (U) ∩ M = U and r(T (U)) = U . Let T be the collection of all pairs (U,V ) of open sets in M such thatconv(V ) ⊂ T (U), where conv(V ) is the closed convex hull of V in E. The family T has the following properties:(i) for any z ∈ M and its neighborhood U in M there is a neighborhood V ⊂ U of z with (U,V ) ∈ T ; (ii) for any(U,V ) ∈ T and open sets U ′,V ′ ⊂ M we have (U ′,V ′) ∈ T provided U ⊂ U ′ and V ′ ⊂ V . By [6, Proposition 2.3](see also [1, Lemma 6.7]), there exists a complete bounded metric � on M such that for every z ∈ M and δ ∈ (0,1)

the pair of open balls (B�(z, δ), (B�(z, δ/8)) belongs to T .Suppose now that A ⊂ Z is closed, ϕ :Z → M and g :A → M , where Z is paracompact and �(g(z),ϕ(z)) <

α(z)/8 for all z ∈ A. Consider the set-valued map Φ :Z → E, Φ(z) = g(z) if z ∈ A and Φ(z) = conv(B�(ϕ(z),α(z)/8))

if z /∈ A. Then Φ is lower semi-continuous and has closed and convex values in E. So, by the Michaelconvex-valued selection theorem [19], Φ has a continuous selection g1. According to the definition of T , everyconv(B�(ϕ(z),α(z)/8)) is contained in r−1(B�(ϕ(z),α(z))). Hence, g = r ◦ g1 is the required extension of g. �

Everywhere below we equip M with a complete metric � satisfying the hypotheses of Lemma 2.1.

Lemma 2.2. Let g ∈ Hi (y) for some y ∈ Y and i � 1. Then there exists a neighborhood Vy of y in Y and δy > 0 suchthat y′ ∈ Vy and �(g1(x), g(x)) < δy for all x ∈ f −1(y′) yields g1 ∈Hi (y

′).

Proof. Since g ∈ Hi (y), g ◦ h−1y :h(f −1(y)) → M is Di -crooked. Hence, h(f −1(y) ∩ g−1(t)) is Di -crooked for

every t ∈ g(f −1(y)). Consequently, h(f −1(y) ∩ g−1(t)) has a neighborhood W(t) in Q which is also Di -crooked.Then U(t) = h−1(W(t)) is a neighborhood of f −1(y) ∩ g−1(t) in X. Hence, we can find neighborhoods Vy(t) andV (t) = B(t,3δ(t)) of y and t in Y and M , respectively, such that G(t,3δ(t)) = f −1(Vy(t))∩ g−1(V (t)) ⊂ U(t) (thiscan be done because f is perfect, so is the map f g :X → Y ×M). Here, B(t,3δ(t)) denotes the open ball in (M,�)

with center t and radius 3δ(t). Next, choose finitely many points {tj : j = 1,2, . . . , k} and a neighborhood Vy of y with

Vy ⊂ ⋂j=k

j=1 Vy(tj ) and f −1(y) ⊂ f −1(Vy) ⊂ ⋃j=k

j=1 G(tj , δ(tj )), and let δy = min{δ(tj ): j = 1, . . . , k}. Let us showthat Vy and δy satisfy the requirement of the lemma. Suppose y′ ∈ Vy and g1 ∈ C(X,M) with �(g1(x), g(x)) < δy

for all x ∈ f −1(y′). Then any x ∈ f −1(y′) is contained in some G(tj , δ(tj )). So, g(x) ∈ B(tj , δ(tj )). It easily fol-lows that g(g−1

1 (g1(x))) ⊂ V (tj ). Hence, g−11 (g1(x))∩f −1(y′) ⊂ g−1(V (tj ))∩f −1(Vy(tj )) ⊂ U(tj ). Consequently,

h(g−11 (g1(x)) ∩ f −1(y′)) ⊂ W(tj ) which implies that h(g−1

1 (g1(x)) ∩ f −1(y′)) is Di -crooked because so is W(tj ).Therefore, g1 ◦ h−1

y′ is a Di -crooked map, i.e., g1 ∈ Hi (y′). This completes the proof. �

Now, we are in a position to show that the sets Hi (Y ) are open in C(X,M).

Proposition 2.3. For any set F ⊂ Y and any i � 1, the set Hi (F ) is open in C(X,M) with respect to the sourcelimitation topology.

Proof. Let F ⊂ Y and g0 ∈ Hi (F ). Then, by Lemma 2.2, for every y ∈ F there exist a neighborhood Vy and apositive δy � 1 such that g ∈ Hi (Vy) provided g|f −1(Vy) is δy -close to g0|f −1(Vy). The family {Vy ∩ Y : y ∈ F }can be supposed to be locally finite in F . Consider the set-valued lower semi-continuous map ϕ :F → (0,1], ϕ(y) =⋃{(0, δz]: y ∈ Vz}. By [23, Theorem 6.2, p. 116], ϕ admits a continuous selection β :F → (0,1]. Let β : Y → (0,1]be a continuous extension of β and α = β ◦ f . It remains only to show that if g ∈ C(X,M) with �(g0(x), g(x)) <


α(x)/8 for all x ∈ X, then g ∈ Hi (F ). So, we take such a g and fix y ∈ F . Then there exists z ∈ F with y ∈ Vz

and α(x) � δz for all x ∈ f −1(y). By Lemma 2.1, we can select a map g′ ∈ C(X,M) coinciding with g on f −1(y)

and satisfying the inequality �(g′(x), g0(x)) � δz for each x ∈ X. According to the choice of Vz, g′ ∈ Hi (y). Hence,g ∈Hi (y) because g|f −1(y) = g′|f −1(y). Therefore, Hi (F ) is open in C(X,M). �2.2. Every Hi (F ) is dense in C(X,M)

In this subsection we suppose M is a free ANR-space equipped with a complete metric � satisfying the hypothesesof Lemma 2.1. We are going to show if F ⊂ Y is closed, then the set Hi (F ) is dense in C(X,M) with respectto the source limitation topology in any of the following cases: (i) F is finite-dimensional; (ii) M is a convex freesubset of a Banach space. In any of these two cases we need to show that B�(g, ε) = {g′ ∈ C(X,M): �(g,g′) < ε}meets Hi (F ) for every g ∈ C(X,M) and every continuous function ε :X → (0,1]. To this end, fix g0 ∈ C(X,M)

and ε ∈ C(X, (0,1]). Consider the set-valued map Φε :Y → C(X,M), Φε(y) = Hi (y) ∩ B�(g0, ε), where C(X,M)

carries the compact open topology.

Lemma 2.4. The map Φε has the following property: If y0 ∈ Y and a compactum K is contained in Φε(y0), then thereexists a neighborhood V (y0) of y0 such that K ⊂ Φε(y) for every y ∈ V (y0).

Proof. Suppose there exists a sequence {yj }j�1 converging to y0 in Y such that K\Φε(yj ) = ∅. Let gj ∈ K\Φε(yj ),j � 1, and P = f −1(Y0), where Y0 = {yj }j�1 ∪{y0}. Since the restriction map πP :C(X,M) → C(P,M) is continu-ous when both C(X,M) and C(P,M) are equipped with the compact open topology, there exists a subsequence {gjk

}of {gj } such that πP (gjk

) converges to πP (g) in C(P,M) for some g ∈ K . Obviously g ∈ Φε(y0) ⊂ Hi (y0). So, wecan apply Lemma 2.2 to find a neighborhood V of y0 in Y and a positive δ > 0 such that y′ ∈ V and �(g(x), g′(x)) < δ

for all x ∈ f −1(y′) implies g′ ∈ Hi (y′). Since πP (gjk

) converges to πP (g) in C(P,M) and the compact open topol-ogy on C(P,M) coincides with the uniform convergence (recall that P is compact), there exists jk with yjk

∈ V and�(g(x), gjk

(x)) < δ for any x ∈ f −1(yjk). Hence, gjk

∈ Hi (yjk). Consequently, gjk

∈ Φε(yjk) which contradicts the

choice of the functions gj . �Lemma 2.5. Every Φε(y) has the following property: If v : Sn → Φε(y) is continuous, where n � 0, then v can beextended to a continuous map u : Bn+1 → Φ64ε(y).

Proof. Let us mention the following property of the function space C(X,M) equipped with the compact open topol-ogy: For any metrizable space Z a map w :Z → C(X,M) is continuous if and only if the map w :Z × X → M ,w(z, x) = w(z)(x), is continuous. Hence, every map v : Sn → Φε(y) generates a continuous map v : Sn × X → M

defined by v(z, x) = v(z)(x) such that �(v(z, x), g0(x)) < ε(x) for all (z, x) ∈ Sn × X.

Claim. Let F ⊂ X be closed and πF :C(X,M) → C(F,M) be the restriction map. Then πF is continuous and openwhen both C(X,M) and C(F,M) are equipped with the source limitation topology.

Suppose G ⊂ C(F,M) is open and πF (g0) ∈ G for some g0 ∈ C(X,M). Then there exists η ∈ C(F, (0,1]) suchthat πF (g) ∈ G provided g ∈ C(X,M) and �(g0(x), g(x)) < η(x) for all x ∈ F . Obviously, B�(g0, η) ⊂ π−1

F (G),where η :X → (0,1] is a continuous extension of η. So, πF is continuous. To show that πF is open, let W ⊂ C(X,M)

be open and πF (g) ∈ πF (W), g ∈ W . Hence, B�(g,α) ⊂ W for some continuous function α :X → (0,1]. Then

O(g) = {q ∈ C(F,M): �

(q(x), g(x)

)< α(x)/8 for all x ∈ F

}

is a neighborhood of πF (g) in C(F,M). According to Lemma 2.1, O(g) ⊂ πF (Hi (y)) which completes the proof ofthe claim.

Now, let πy be the restriction map πf −1(y). Then, by the above claim, πy(Hi (y)) is open in C(f −1(y),M) withrespect to the source limitation topology. Since f −1(y) is compact, the source limitation, the compact open andthe uniform convergence topologies on C(f −1(y),M) coincide. On the other hand, πy is continuous when bothC(X,M) and C(f −1(y),M) carry the compact open topology. Hence, πy(v(Sn)) is a compact subset of πy(Hi (y))

and πy(Hi (y)) is open in C(f −1(y),M), where C(f −1(y),M) carries the uniform convergence topology generated


by the metric �. Consequently, there is δ1 > 0 such that if β : Sn → C(f −1(y),M) and �(β(z, x), v(z, x)) < δ1 for all(z, x) ∈ S

n × f −1(y), then β(Sn) ⊂ πy(Hi (y)).Define δ2 = inf{ε(x) − �(v(z, x), g0(x)): (z, x) ∈ S

n × f −1(y)}. Obviously, δ2 > 0. According to Lemma 2.1,there exists a continuous extension v1 : Bn+1 × f −1(y) → M of the map v|(Sn × f −1(y)) with �(v1(z, x), g0(x)) <

8ε(x) for all (z, x) ∈ Bn+1 × f −1(y). Since B

n+1 × f −1(y) is compact and M is a free space, we can finda Bing map v2 : Bn+1 × f −1(y) → M with �(v2(z, x), v1(z, x)) < δ/8 for all (z, x) ∈ B

n+1 × f −1(y), whereδ = min{δ1, δ2, δ3} and δ3 = inf{8ε(x) − �(v1(z, x), g0(x)): (z, x) ∈ B

n+1 × f −1(y)}. Therefore, we have a mapv2 : Bn+1 → C(f −1(y),M). The choice of δ3 implies

�(v2(z, x), g0(x)

)< 8ε(x) (1)

for all (z, x) ∈ Bn+1 ×f −1(y). Moreover, since v2 is a Bing map, so are the maps v2(z) :f −1(y) → M , z ∈ B

n+1. Onthe other hand, by Lemma 2.1 and (1), every map v2(z) can be extended to a map from X into M . Therefore,

v2(Bn+1) ⊂ πy

(Hi (y)

). (2)

Representing the ball Bn+1 as a cone with a base S

n and a vertex z0, we can consider v2 as a homotopy fromS

n × f −1(y) × [0,1] into M between the maps v2|(Sn × f −1(y) × {0}) and v2|({z0} × f −1(y)). Observe also that�(v2(z, x,0), v(z, x)) < δ/8 for any (z, x) ∈ S

n × f −1(y). Let E be the Banach space from the proof of Lemma 2.1and ϕ : Sn × f −1(y) × [0,1] → E be the set valued map defined by

ϕ(z, x, t) =

⎧⎪⎨

⎪⎩

v(z, x) if t = 0;v2(z, x,0) if t = 1;conv(B�(v(z, x), δ/8)) if t ∈ (0,1).

By the convex-valued selection theorem of Michael, ϕ admits a continuous selection s. It follows from the proof ofLemma 2.1 that v3 = r ◦ s is a map from S

n × f −1(y) × [0,1] into M such that v3(z, x,0) = v(z, x), v3(z, x,1) =v2(z, x,0) and �(v3(z, x, t), v(z, x)) < δ for every (z, x, t) ∈ S

n × f −1(y) × [0,1]. Since δ < min{δ1, δ2}, for any(z, x, t) ∈ S

n × f −1(y) × [0,1] we have

�(v3(z, x, t), v(z, x)

)< δ1, (3)

and

�(v3(z, x, t), g0(x)

)< ε(x). (4)

Therefore, v3 is a homotopy connecting the maps v and v2|(Sn ×f −1(y)×{0}), while v2 is a homotopy connectingthe maps v2|(Sn ×f −1(y)×{0}) and v2|({z0}×f −1(y)). Combining these two homotopies, we obtain a map u1 : Sn ×f −1(y) × [0,1] → M such that u1(z, x,0) = v(z, x), u1(z, x,1) = v2(z0, x) and �(u1(z, x, t), g0(x)) < 8ε(x) forall (z, x, t) ∈ S

n × f −1(y) × [0,1]. Obviously, u1 can also be considered as a map from Bn+1 × f −1(y) into M

such that u1|(Sn × f −1(y)) = v and �(u1(z, x), g0(x)) < 8ε(x), (z, x) ∈ Bn+1 × f −1(y). Now consider the map

u2 : (Bn+1 ×f −1(y))∪(Sn×X) → M with u2|(Bn+1 ×f −1(y)) = u1 and u2|(Sn×X) = v. Finally, using Lemma 2.1,we extend u2 to a map u : Bn+1 × X → M such that

�(u(z, x), g0(x)

)< 64ε(x) (5)

for any (z, x) ∈ Bn+1 × X. Then u : Bn+1 → C(X,M) extends the map v. Moreover, (2), (3) and the choice of δ1

implies that u(Bn+1) ⊂ Hi (y). On the other hand, (5) yields u(Bn+1) ⊂ B�(g0,64ε). Hence, u(Bn+1) ⊂ Φ64ε(y). �Next proposition completes the proof of Theorem 1.1 in the case Y is strongly countable-dimensional.

Proposition 2.6. The sets Hi (F ), i � 1, are dense in C(X,M) with respect to the source limitation topology for anyclosed finite-dimensional set F ⊂ Y .

Proof. If F ⊂ Y is closed and dimF � n, we consider the sequence of the set-valued maps Φj :F → C(X,M),j = 0, . . . , n, defined by Φj(y) = Φε/82(n−j)+1(y). Obviously, Φ0(y) ⊂ Φ1(y) ⊂ · · · ⊂ Φn(y) = Φε/8(y). Accord-

ing to Lemma 2.5, every map from Sn into Φj(y) can be extended to a map from B

n+1 into Φj+1(y), where


j = 0,1, . . . , n − 1 and y ∈ F . Moreover, by Lemma 2.4, any Φj(y) has the following property: if K ⊂ Φj(y) iscompact, then there exists a neighborhood Vy of y in Y such that K ⊂ Φj(z) for all z ∈ Vy . So, we may apply [9, The-orem 3.1] to find a continuous selection θ :F → C(X,M) of Φn. Hence, θ(y) ∈ Φε/8(y) for all y ∈ F . Now, define amap g :f −1(F ) → M , g(x) = θ(f (x))(x). To show that g is continuous, fix a sequence {xk} ⊂ f −1(F ) converging tosome x0 ∈ f −1(F ) and let yk = f (xk), k � 0. Since C(X,M) carries the compact open topology, the sequence {θ(yk)}restricted to any compact set P ⊂ f −1(F ) converges uniformly to θ(y0)|P . Letting P = f −1({yk}k�0), one can easilyshow that {g(xk} converges to g(x0). Hence g is continuous and �(g(x), g0(x)) < ε(x)/8 for all x ∈ f −1(F ). Then,by Lemma 2.1, g can be extended to a continuous map g :X → M with �(g(x), g0(x)) < ε(x), x ∈ X. It follows fromthe definition of g that g|f −1(y) = θ(y)|f −1(y) for every y ∈ F . Since θ(y) ∈ Hi (y), we have g ∈ Hi (F ). Hence,B�(g0, ε) ∩Hi (F ) = ∅. �

We now turn to the proof of Theorem 1.1 in the case Y is a C-space and M a closed convex free subset of aBanach space E. Let � be the metric on M inherited from the norm of E and Ψε :Y → C(X,M) be the set-valuedmap Ψε(y) = B�(g0, ε) ∩Hi (y), where C(X,M) is equipped with the compact open topology and

B�(g0, ε) = {g ∈ C(X,M): �

(g0(x), g(x)

)� ε(x) for all x ∈ X

}.

Lemma 2.7. Ψε has the following property: Every map v : Sn → Ψε(y), n � 0, can be extended to a map u : Bn+1 →Ψε(y).

Proof. All function spaces in this proof are equipped with the compact open topology. Let πy :C(X,M) →C(f −1(y),M) be the restriction map and P(y) = B�(g0, ε, y)\πy(Hi (y)), where B�(g0, ε, y) is the set

{g ∈ C

(f −1(y),M

): �

(g0(x), g(x)

)� ε(x) for all x ∈ f −1(y)

}.

According to the proof of Lemma 2.5, P(y) is closed in B�(g0, ε, y).We are going to show that P(y) is a Z-set in B�(g0, ε, y), i.e., every map w :K → B�(g0, ε, y), K is any compact,

can be approximated by a map w1 :K → B�(g0, ε, y)\P(y) = B�(g0, ε, y) ∩ πy(Hi (y)). To this end, fix δ > 0 andlet w :K × f −1(y) → M be the map generated by w. So, �(w(z, x), g0(x)) � ε(x) for all (z, x) ∈ K × f −1(y).Since f −1(y) is compact, there exists λ ∈ (0,1) such that λmax{ε(x): x ∈ f −1(y)} < δ/2. Define the map w1 :K ×f −1(y) → M by w1(z, x) = (1 − λ)w(z, x) + λg0(x). Then, for all (z, x) ∈ K × f −1(y) we have

�(w1(z, x),w(z, x)

)� λε(x) < δ/2

and

�(w1(z, x), g0(x)

)� (1 − λ)ε(x) < ε(x).

Since M is a free space, there exists a Bing map w2 :K × f −1(y) → M which is δ1-close to w1, where δ1 =min{λε(x): x ∈ f −1(y)}. Hence, �(w2(z, x), g0(x)) � ε(x) and �(w2(z, x),w(z, x)) < δ, (z, x) ∈ K × f −1(y). Thelast two inequalities imply that the map w2 :K → C(f −1(y),M) is δ-close to w and w2(K) ⊂ B�(g0, ε, y). More-over, every map w2(z) :f −1(y) → M , z ∈ K , can be extended to a map from X to M because M is a closed convexsubset of E. Since w2 is a Bing map, so are the maps w2(z), z ∈ K . Hence, w2(K) ⊂ πy(Hi (y)). So, P(y) is a Z-setin B�(g0, ε, y).

Now we can complete the proof of the lemma. For every map v : Sn → Ψε(y) the composition πy ◦ v is a mapfrom S

n into B�(g0, ε, y) ∩ πy(Hi (y)). Since P(y) is a Z-set in the convex set B�(g0, ε, y), by [25, Proposi-tion 6.3], there exists a map v1 : Bn+1 → B�(g0, ε, y) ∩ πy(Hi (y)) extending πy ◦ v. Consider the map v2 :A → M ,where A = (Bn+1 × f −1(y)) ∪ (Sn × X), defined by v2|(Bn+1 × f −1(y)) = v1 and v2|(Sn × X) = v. Next, takea selection u : Bn+1 × X → M for the set-valued map φ : Bn+1 × X → M , φ(z, x) = v2(z, x) if (z, x) ∈ A andφ(z, x) = B�(g0(x), ε(x)) if (z, x) /∈ A. Such u exists by Michael’s convex-valued selection theorem. Obviously u

extends v2 and �(u(z, x), g0(x)) � ε(x) for every (z, x) ∈ Bn+1 × X. Finally, observe that u is the required extension

of v. �We can finally finish the proof of Theorem 1.1.


Proposition 2.8. Suppose Y is a C-space and M a closed convex free subset of a Banach space E. Then the setsHi (F ), i � 1, are dense in C(X,M) with respect to the source limitation topology for any closed set F ⊂ Y .

Proof. For every ε, i and g0 ∈ C(X,M) consider the set-valued map Ψε :F → C(X,M). It follows from the proof ofLemma 2.4 that if K ⊂ Ψε(y0) for some compactum K and y0 ∈ F , then y0 admits a neighborhood V with K ⊂ Ψε(y)

for all y ∈ V . Moreover, according to Lemma 2.7, every image Ψε(y) is aspherical, i.e., any map from Sn into Ψε(y),

n � 0, can be extended to a map from Bn+1 to Ψε(y). Then, by the Uspenskij selection theorem [25, Theorem 1.3], Ψε

admits a continuous selection θ :F → C(X,M). Repeating the arguments from the proof of Proposition 2.6, we obtaina map g :f −1(F ) → M such that �(g(x), g0(x)) � ε(x) for every x ∈ f −1(F ) and g|f −1(y) = θ(y)|f −1(y), y ∈ F .Applying once more the Michael convex-valued selection theorem for the set-valued map ϑ :X → M , ϑ(x) = g(x) ifx ∈ F and ϑ(x) = B�(g0(x), ε(x)) if x /∈ F , we obtain a selection g for ϑ . Obviously, g extends g and g ∈ B�(g0, ε).Since θ(y) ∈ Hi (y) for all y ∈ F , we have g ∈ B�(g0, ε) ∩Hi (F ). Hence, Hi (F ) is dense in C(X,M). �3. Proof of Theorem 1.2

Let f :X → Y be a perfect map between with X,Y metrizable and Y is a C-space. We say that g ∈ C(X, I) is anf -Krasinkiewicz map if the restrictions g|f −1(y) are Krasinkiewicz maps for every y ∈ Y . We are going to show thatall f -Krasinkiewicz maps contains a dense Gδ-subset of C(X, I) with respect to the source limitation topology. Tothis end, we use an idea of Levin–Lewis from the proof of [17, Proposition 3.3]. We fix a metric d on X and for everyy ∈ Y , natural number i and rational numbers p,q with 0 � p < q � 1 let Ki (p, q, y) = C(X, I)\Li (p, q, y), whereLi (p, q, y) is the set of all g ∈ C(X, I) satisfying the following condition: there exists a continuum F ⊂ f −1(y) suchthat [p,q] ⊂ g(F ) and for every t ∈ [p,q] there exists a component C of g−1(t) ∩ f −1(y) and a point x ∈ C withC ∩ F = ∅ and d(x,F ) � 1/i. For a set H ⊂ Y we denote by Ki (p, q,H) the intersection of all Ki (p, q, y), y ∈ H .It is easily seen that

K =⋂{

Ki (p, q,Y ): i � 1 and p,q rationals}

consists of f -Krasinkiewicz maps. Moreover every g ∈ K has the following property:

(∗) For every y ∈ Y and every continuum F ⊂ f −1(y) with g(F ) being not a singleton there exists a dense subsetD ⊂ g(F ) such that g−1(t) ∩ F is the union of components of g−1(t) ∩ f −1(y) for every t ∈ D.

Therefore, it suffices to show that each set Ki (p, q,Y ) is open and dense in C(X, I) with respect to the sourcelimitation topology.

Proposition 3.1. For every closed H ⊂ Y , the set Ki (p, q,H) is open in C(X, I).

Proof. We are going first to prove the following

Claim. If g ∈Ki (p, q, y) for some y ∈ Y , then there exists a neighborhood Vy of y in Y and δy > 0 such that y′ ∈ Vy

and |g′(x) − g(x)| < δy for all x ∈ f −1(y′) yields g′ ∈ Ki (p, q, y′).

Indeed, otherwise we can find a local base of sequences {Vn} of neighborhoods of y in Y , points yn ∈ Vn andfunctions gn ∈ C(X, I) such that |gn(x) − g(x)| < 1/n for all x ∈ f −1(yn) but gn /∈Ki (p, q, yn), n � 1. Hence, gn ∈Li (p, q, yn) for all n. Consequently, for every n there is a continuum Fn ⊂ f −1(yn) such that [p,q] ⊂ gn(Fn) andfor any t ∈ [p,q] there exists a component Cn(t) of g−1

n (t) ∩ f −1(yn) and a point xn(t) ∈ Cn(t) with Cn(t) ∩ Fn = ∅and d(xn(t),Fn) � 1/i. Then all Fn are contained in the compact set P = f −1({yn} ∪ {y}). Passing to subsequences,we may suppose that {Fn}, considered as a sequence in the space expC(P ) of all subcontinua of P equipped withthe Hausdorff metric generated by d , converges to a continuum F . It follows that F ⊂ f −1(y) and [p,q] ⊂ g(F ).Since g ∈ Ki (p, q, y), there exists t0 ∈ [p,q] such that d(C,F ) < 1/i for every continuum C ⊂ g−1(t0) ∩ f −1(y)

with C ∩ F = ∅. Next, consider the sequences {Cn(t0)} and {xn(t0)}. Passing again to subsequences, we may assumethat {Cn(t0)} converges in expC(P ) to a continuum C0 and {xn(t0)} converges in P to a point x0. It is easily seen that


C0 ⊂ g−1(t0) ∩ f −1(y), C0 ∩ F = ∅, x0 ∈ C0 and d(x0,F ) � 1/i. On the other hand, according to the choice of t0we have d(C0,F ) < 1/i. This contradiction completes the proof of the claim.

Now, repeating the arguments from the proof of Proposition 2.3 and applying the above claim instead ofLemma 2.2, we can show that every Ki (p, q,H), H ⊂ Y is closed, is open in C(X, I). �Proposition 3.2. Every Ki (p, q,Y ) is dense in C(X, I) with respect to the source limitation topology.

Proof. All function spaces in this proof are considered with the compact open topology. For every ε ∈ C(X, (0,1])and g0 ∈ C(X, I) consider the set-valued map Ωε :Y → I defined by Ωε(y) = B(g0, ε) ∩Ki (p, q, y), where

B(g0, ε) = {g ∈ C(X, I):

∣∣g0(x) − g(x)∣∣ � ε(x) for all x ∈ X

}.

Claim 1. Every map v : Sn → Ωε(y), n � 0, can be extended to a map u : Bn+1 → Ωε(y).

Following the notations from the proof of Lemma 2.7, let P(y) = B(g0, ε, y)\πy(Ki (p, q, y)), where B(g0, ε, y)

is the set{g ∈ C

(f −1(y), I

):

∣∣g0(x) − g(x)∣∣ � ε(x) for all x ∈ f −1(y)

}.

One can easily show that P(y) is closed in B(g0, ε, y). As in Lemma 2.7, we show first that P(y) is a Z-set inB(g0, ε, y). The only difference is that we take now w2 :K × f −1(y) → I to be a Krasinkiewicz map on K × f −1(y)

satisfying the following condition: For every continuum F :K × f −1(y) with w2(F ) being not a singleton there is adense subset D ⊂ w2(F ) such that w−1

2 (t) ∩ F is the union of components of w−12 (t) for every t ∈ D. This can be

done by the Levin–Lewis result [17, Proposition 3.3] stating that all maps with the above property form a dense subsetof C(K × f −1(y), I). Moreover, every map w2(z) :f −1(y) → I, z ∈ K , belongs to πy(Ki (p, q, y)). Then, followingthe arguments from the proof of Lemma 2.7, we complete the proof of the claim.

Claim 2. The map Ωε has the following property: If y0 ∈ Y and a compactum K is contained in Ωε(y0), then thereexists a neighborhood V of y0 in Y such that K ⊂ Ωε(y) for every y ∈ V .

The proof of this claim is the same as that one of Lemma 2.4. The only difference is that we need now to apply theclaim from Proposition 3.1 instead of Lemma 2.2.

Finally, because of Claims 1 and 2, we can apply the Uspenskij selection theorem [25, Theorem 3.1] to finda continuous selection θ :Y → C(X, I) for the map Ωε . Then the map g :X → I, g(x) = θ(f (x))(x) is continu-ous and g ∈ B(g0, ε). Moreover, since g|f −1(y) = θ(y)|f −1(y) for every y ∈ Y , we have g ∈ Ki (p, q,Y ). Hence,Ki (p, q,Y ) is dense in C(X, I). �4. Some remarks and questions

As we mention in the introduction, there are free spaces which are not necessarily ANR. Here is a version ofTheorem 1.1 for such spaces.

Theorem 4.1. Let M be a completely metrizable LCn free space. Then, for every perfect map f :X → Y with dimY +dimX � n + 1, the function space C(X,M) contains a dense Gδ-set of maps g such that all restrictions g|f −1(y),y ∈ Y , are Bing maps.

Proof. We need the following version of Lemma 2.1, see [2].

Lemma 4.2. Every completely metrizable LCn-space M admits a complete bounded metric � generating its topologyand satisfying the following condition: Let Z be a paracompact space with dimZ � n + 1, A ⊂ Z a closed set andϕ :Z → M a map. Then, for every function α :Z → (0,1] and every map g :A → M with �(g(z),ϕ(z)) < α(z)/8 forall z ∈ A, there exists a map g :Z → M extending g such that �(g(z), ϕ(z)) < α(z) for all z ∈ Z.


Following the proof of Theorem 1.1 (the case M is a free ANR-space and Y strongly countable-dimensional) andusing Lemma 4.2 instead of Lemma 2.1, we can show that the sets Hi (Y ) are open in C(X,M) with respect to thesource limitation topology.

To show the density of the sets Hi (Y ) in C(X,M), let dimY = k. Then the arguments from Lemma 2.5 im-ply that the map Φε has the following property: every map v : Sk−1 → Φε(y) admits a continuous extensionu : Bk → Φ64ε(y), y ∈ Y . Next, we consider the maps Φj :Y → C(X,M), Φj(y) = Φε/82(k−j) (y), j = 0,1, . . . , k,and use [9, Theorem 3.1] to find a continuous selection θ :Y → C(X,M) for Φk . Finally, the map g :X → M definedby g(x) = θ(f (x))(x) satisfies the conditions �(g0(x), g(x)) < ε(x), x ∈ X, and g ∈ Hi (y) for every y ∈ Y (seeProposition 2.6). �

As an application of Theorem 1.1, we establish that some maps have Bula’s property. Following Kato and Levin[10], a map f :X → Y is said to have the Bula property if there exist two closed disjoint subsets F0 and F1 of X suchthat f (F0) = f (F1) = Y . Bula [4] has shown that every open map f from a compact Hausdorff space onto a finite-dimensional metric space has this property provided all fibers of f are dense in themselves. Gutev [8] generalizedBula’s result to the case Y is countable-dimensional. Recently, Levin and Rogers [18] obtained a further generaliza-tion with Y being a C-space. The question whether the Levin–Rogers result remains true for perfect maps betweenmetrizable spaces is still open [11] (if Y is strongly infinite-dimensional, this is not true, see [5] and [18]). Here, weprovide a partial answer to this question:

Theorem 4.3. Let f :X → Y be a perfect open map between metrizable spaces such that Y is a C-space and no fiberof f is a Bing space. Then f has the Bula property.

Proof. We follow the arguments from the Kato–Levin proof of [10, Theorem 2.2]. According to Theorem 1.1, thereexists g :X → I such that every restriction g|f −1(y) ia a Bing map. Since no fiber f −1(y) is a Bing space, eachg(f −1(y)) contains at least two different points. Let ay = min{g(x): x ∈ f −1(y)}, by = max{g(x): x ∈ f −1(y)}and hy : [ay, by] → [0,1] be the linear transformation with hy(ay) = 0 and hy(by) = 1. Then the map h :X → [0,1],h(x) = hy(g(x)) for x ∈ f −1(y), is continuous because f is both closed and open. Moreover, 0,1 ∈ h(f −1(y)),y ∈ Y . Obviously, F0 = h−1(0) and F1 = h−1(1) are closed disjoint subsets of X with f (F0) = f (F1) = Y . �

Most probably Theorem 1.1 remains true when M is a completely metrizable ANR-free space and Y is a C-space.The validity of this more general version of Theorem 1.1 is reduced to the existence of an appropriate selectiontheorem for C-spaces. The Uspenskij theorem does not work in this case because we do not have a single set-valuedmap Φ :Y → C(X,M) with aspherical images such that Φ(y) ⊂ Hi (y) for every y ∈ Y . But we can construct adecreasing sequence of maps Φn :Y → C(X,M), n = 0,1, . . . , such that each pair (Φn+1(y),Φn(y)), n � 0 andy ∈ Y , is UV n (i.e., every map v : Sn → Φn+1(y) can be extended to a map u : Bn+1 → Φn(y)) and each Φn isstrongly lower semi-continuous (i.e., if K ⊂ Φn(y0) for some compact K , then there exists a neighborhood V of y0such that K ⊂ Φn(y) for any y ∈ V ). Having in mind this observation, we can ask the following question.

Question. Let X be a C-space, Y a Tychonoff space and Φn :X → Y , n � 0, a decreasing sequence of strongly lowersemi-continuous maps such that, for any n and x ∈ X, the pair (Φn+1(x),Φn(x)) is UV n. Does Φ0 admit a continuousselection? What about if Y is a completely metrizable ANR or each Φn+1(x) is contractible in Φn(x)?

Note added in proof

In a recent paper with Gutev we provided a positive answer to the question about Bula’s property discussed in thissection. We proved that Theorem 4.3 is true if Y is a paracompact C-space, X a Tychonoff space and f :X → anopen continuous surjection such that all fibers of f are infinite and C∗-embedded in X. Moreover, Gutev establisheda negative answer to the question formulated at the end of the paper.

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Continuous selections avoiding extreme points

Takamitsu Yamauchi

Department of Mathematics, Shimane University, Matsue 690-8504, Japan

Received 6 August 2006; received in revised form 5 February 2007; accepted 5 February 2007

Abstract

It is shown that if X is a countably paracompact collectionwise normal space, Y is a Banach space and ϕ : X → 2Y is a lowersemicontinuous mapping such that ϕ(x) is Y or a compact convex subset with Cardϕ(x) > 1 for each x ∈ X, then ϕ admits acontinuous selection f : X → Y such that f (x) is not an extreme point of ϕ(x) for each x ∈ X. This is an affirmative answerto the problem posed by V. Gutev, H. Ohta and K. Yamazaki [V. Gutev, H. Ohta and K. Yamazaki, Selections and sandwich-likeproperties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55 (2003) 499–521].© 2007 Elsevier B.V. All rights reserved.

MSC: 54C60; 54C65; 54D15; 54D20

Keywords: Set-valued mapping; Selection; Extreme point; Countable paracompactness

1. Introduction

Throughout this paper, all spaces are assumed to be T1. Let X be a space and (Y,‖ · ‖) a Banach space. By 2Y ,Fc(Y ) and C′

c(Y ) we denote the set of all nonempty subsets of Y , the set of all nonempty closed convex subsets of Y

and the set consisting of Y and all nonempty compact convex subsets of Y , respectively. For a mapping ϕ : X → 2Y ,a mapping f : X → Y is called a selection if f (x) ∈ ϕ(x) for each x ∈ X. E. Michael [9, Theorem 3.1′′′] characterizedperfectly normal spaces by means of continuous selections avoiding supporting set. Our concern of this paper is toobtain characterizations in terms of continuous selections avoiding extreme points.

For a convex subset K of Y , a point y ∈ K is called an extreme point if every open line segment containing y is notcontained in K , that is, y1 = y2 = y whenever y1, y2 ∈ K , y = δy1 + (1 − δ)y2 and 0 < δ < 1. For a convex subset K

of Y , the weak convex interior wci(K) of K [5] is the set of all non-extreme points of K , that is,

wci(K) = {y ∈ K | y = δy1 + (1 − δ)y2 for some y1, y2 ∈ K \ {y} and 0 < δ < 1

}.

For K ∈ Fc(Y ), let I (K) be the set of all points of K which are not in any supporting set of K (see [9, p. 372]). Thenwe have I (K) ⊂ wci(K) for every K ∈Fc(Y ) of cardinality >1. The converse inclusion does not hold in general. Forexample, for the convex hull K of three points v0, v1, v2 in Y with {v1 − v0, v2 − v0} linearly independent, we haveI (K) = {y ∈ K | y = ∑2

i=0 δivi , 0 < δi < 1 and∑2

i=0 δi = 1} � K \ {v0, v1, v2} = wci(K).



T. Yamauchi / Topology and its Applications 155 (2008) 916–922 917

By w(Y) we denote the weight of a space Y . A Hausdorff space X is called countably paracompact if everycountable open cover of X is refined by a locally finite open cover of X. By a cardinal number, we mean an initialordinal number. The cardinality of a set S is denoted by CardS. For an infinite cardinal number λ, a T1-space X iscalled λ-collectionwise normal if for every discrete collection {Fα | α ∈ A} of closed subsets of X with CardA � λ,there exists a disjoint collection {Gα | α ∈ A} of open subsets of X such that Fα ⊂ Gα for each α ∈ A. A mappingϕ : X → 2Y is called lower semicontinuous (l.s.c. for short) if for every open subset V of Y , the set ϕ−1[V ] = {x ∈X | ϕ(x) ∩ V �= ∅} is open in X. Let R be the space of real numbers with the usual topology. The space c0(λ) isthe Banach space consisting of functions s : D(λ) → R, where D(λ) is a set with CardD(λ) = λ, such that for eachε > 0 the set {α ∈ D(λ) | |s(α)| � ε} is finite, where the linear operations are defined pointwise and ‖s‖ = sup{|s(α)| |α ∈ D(λ)} for each s ∈ c0(λ). In order to connect insertion theorems with selection theorems, V. Gutev, H. Ohtaand K. Yamazaki [5] introduced lower and upper semicontinuity of a mapping to the Banach space c0(λ) and, withthe aid of these concepts, they proved sandwich-like characterizations of paracompact-like properties. Moreover, theyintroduced generalized c0(λ)-spaces for Banach spaces and established the following theorem [5, Theorem 4.5], whichis an extension of the characterization of normal countably paracompact spaces due to C.H. Dowker [2, Theorem 4]and M. Katetov [6, Theorem 2].

Theorem 1. (See Gutev, Ohta and Yamazaki [5].) For a T1-space X, the following statements are equivalent.

(a) X is countably paracompact and λ-collectionwise normal.(b) For every generalized c0(λ)-space Y and every l.s.c. mapping ϕ : X → C′

c(Y ) with Cardϕ(x) > 1 for each x ∈ X,there exists a continuous selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X.

(c) For every closed subset A of X and every two mappings g,h : A → c0(λ) such that g is upper semicontinuous, h

is lower semicontinuous and g(x) < h(x) for each x ∈ A, there exists a continuous mapping f : X → c0(λ) suchthat g(x) < f (x) < h(x) for each x ∈ A.

Concerning this theorem they posed the following problem [5, Problem 4.7]: Does condition (b) of Theorem 1remain true by replacing “every generalized c0(λ)-space Y ” with “every Banach space Y with w(Y) � λ”? The mainpurpose of this paper is to give an affirmative answer to this problem. In Section 2, we prove the following theorem.

Theorem 2. A T1-space X is countably paracompact and λ-collectionwise normal if and only if for every Banachspace Y with w(Y) � λ and every l.s.c. mapping ϕ : X → C′

c(Y ) with Cardϕ(x) > 1 for each x ∈ X, there exists acontinuous selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X.

Note that, in the proof of the “only if” part of Theorem 2, we does not use the following Dowker–Katetov’sinsertion theorem, which was applied in the proof of (a) ⇒ (b) of Theorem 1. Thus Theorem 2 provides the followingDowker–Katetov’s insertion theorem.

Theorem 3. (See Dowker [2], Katetov [6].) If X is a normal countably paracompact space, then for every uppersemicontinuous function g : X → R and every lower semicontinuous function h : X → R with g(x) < h(x) for eachx ∈ X, there exists a continuous function f : X → R such that g(x) < f (x) < h(x) for each x ∈ X.

In Section 3, we obtain some selection theorems including the following variation of Michael’s convex-valuedselection theorem [9, Theorems 3.2′′].

Theorem 4. A T1-space X is paracompact if and only if for every Banach space Y and every l.s.c. mapping ϕ :X → Fc(Y ) such that Cardϕ(x) > 1 for each x ∈ X, there exists a continuous selection f : X → Y of ϕ such thatf (x) ∈ wci(ϕ(x)) for each x ∈ X.

Let N denote the set of all positive integers. The closure (resp., boundary) of a subset S of a space is denoted byClS (resp., BdS). For y ∈ Y and r > 0, the open ball {y′ ∈ Y | ‖y − y′‖ < r} is denoted by B(y, r). For A,B ∈ 2Y , letdist(A,B) = inf{‖y1 − y2‖ | y1 ∈ A,y2 ∈ B} and diamA = sup{‖y1 − y2‖ | y1, y2 ∈ A}. For a collection V of subsets

918 T. Yamauchi / Topology and its Applications 155 (2008) 916–922

of Y , put meshV = sup{diamV | V ∈ V \ {∅}}. By convA we denote the convex hull of a subset A of Y . For undefinedterminology, we refer to [4].

2. Proof of Theorem 2

To prove Theorem 2, we use the following two lemmas.

Lemma 5. Let X be a λ-collectionwise normal space, Y a Banach space with w(Y) � λ, ϕ : X → C′c(Y ) an l.s.c.

mapping and V an open convex subset of Y . If ϕ(x) ∩ V �= ∅ for each x ∈ X, then the mapping ψ : X → C′c(ClV )

defined by ψ(x) = Cl(ϕ(x) ∩ V ) for each x ∈ X admits a continuous selection.

Proof. Suppose that ϕ(x) ∩ V �= ∅ for each x ∈ X and let ψ : X → C′c(ClV ) be the mapping defined as above. Then,

due to [9, Propositions 2.3 and 2.4], ψ is l.s.c. By C(Y ) and Cc(Y ), we denote the set of all nonempty compact subsetsof Y and the set of all nonempty compact convex subsets of Y , respectively. Since ClV is completely metrizable,due to [10, Theorem 4.4], there exists an l.s.c. mapping θ : X → C(Y ) such that θ(x) ⊂ ψ(x) for each x ∈ X. ThenCl(conv θ(x)) ⊂ ψ(x) for each x ∈ X. By virtue of Mazur’s compactness theorem (cf. [7, 2.8.15]), Cl(conv θ(x)) iscompact for each x ∈ X. Moreover, the mapping φ : X → Cc(Y ) defined by φ(x) = Cl(conv θ(x)) for each x ∈ X

is l.s.c. Thus, by Michael’s selection theorem [9, Theorem 3.2′] (see also [1], [10, Theorem 4.2]), there exists acontinuous selection f : X → Y of φ. Then f is a continuous selection of ψ . �

The following lemma was essentially proved by V. Gutev, H. Ohta and K. Yamazaki (see the last paragraph in theproof of (1) ⇒ (2) of [5, Theorem 4.5]).

Lemma 6. (See Gutev, Ohta and Yamazaki [5].) Let X be a normal space, Y a Banach space and ϕ : X → Fc(Y ) amapping. If there exist a locally finite open cover {Sα | α ∈ A} of X and a collection of {f 1

α , f 2α | α ∈ A} of continuous

selections f 1α , f 2

α : Sα → Y of ϕ|Sα with f 1α (x) �= f 2

α (x) for each x ∈ Sα and α ∈ A, then there exists a continuousselection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X.

Now we prove Theorem 2.

Proof of Theorem 2. The “if” part follows from Theorem 1 since generalized c0(λ)-spaces are Banach spaces withw(Y) � λ [5, Definition 4.4]. To prove the “only if” part, let X be a countably paracompact λ-collectionwise normalspace, (Y,‖ · ‖) a Banach space with w(Y) � λ and ϕ : X → C′

c(Y ) an l.s.c. mapping with Cardϕ(x) > 1 for eachx ∈ X. We first construct a locally finite open cover {Sα | α ∈ A} of X and a collection {V k

α | α ∈ A, k = 1,2} of openconvex subsets on Y such that ClV 1

α ∩ ClV 2α = ∅ and ϕ(x) ∩ V k

α �= ∅ for each x ∈ ClSα and k = 1,2.For each i ∈ N, take a locally finite open cover Vi of Y such that meshVi < 1/2i+1. Since Vi is locally finite,

CardVi � λ for each i ∈ N. Put Oi = {x ∈ X | diamϕ(x) > 1/2i−1} for each i ∈ N. Since ϕ is l.s.c. and Cardϕ(x) > 1for each x ∈ X, the collection {Oi | i ∈ N} is a countable open cover of X. There exists a locally finite open cover{Gi | i ∈ N} of X such that ClGi ⊂ Oi for each i ∈ N.

Let i ∈ N with Gi �= ∅. For each x ∈ ClGi , there exist V 1x ,V 2

x ∈ Vi such that Cl(convV 1x ) ∩ Cl(convV 2

x ) = ∅and ϕ(x) ∩ V k

x �= ∅ for k = 1,2. Indeed, take y1x , y2

x ∈ ϕ(x) such that ‖y1x − y2

x‖ > 1/2i−1. Choose V 1x ,V 2

x ∈ Vi

satisfying ykx ∈ V k

x for each k = 1,2. Since diam convV kx = diamV k

x , we have dist(convV 1x , convV 2

x ) � 1/2i−1 −diamV 1

x − diamV 2x � 1/2i , and hence Cl(convV 1

x )∩ Cl(convV 2x ) = ∅. Put Ux = ϕ−1[V 1

x ]∩ϕ−1[V 2x ]∩ ClGi and set

Ui = {Ux | x ∈ ClGi}. Then Ui is an open cover of ClGi such that CardUi � Card(Vi × Vi ) � λ. Take an arbitraryUi ∈ Ui . Since ϕ(x) is compact for each x ∈ ClGi \ Ui and Vi is locally finite, the collection {U \ Ui | U ∈ Ui} isa point-finite open cover of ClGi \ Ui . Because ClGi is λ-collectionwise normal, by virtue of [8, Theorem 2] and[3, Lemma 1], there exists a locally finite open cover {HU | U ∈ Ui} of ClGi such that HU \ Ui ⊂ U \ Ui for eachU ∈ Ui . Put WU = HU ∩ U ∩ Gi for each U ∈ Ui \ {Ui} and WUi

= Ui ∩ Gi . Then {WU | U ∈ Ui} is a locally finite(in X) collection of open subsets of X such that WU ⊂ U for each U ∈ Ui and Gi = ⋃

U∈UiWU . For each U ∈ Ui ,

choose xU ∈ ClGi so that U = UxUand let V k

U = convV kxU

for k = 1,2. Then V kU is an open convex subset of Y

for each k = 1,2.


Put Ui = ∅ for i ∈ N with Gi = ∅ and set A = ⋃i∈N Ui . Since the collection {Wα | α ∈ A} is a locally finite open

cover of X, there is an open cover {Sα | α ∈ A} of X such that ClSα ⊂ Wα for each α ∈ A. Then {Sα | α ∈ A} and{V k

α | α ∈ A, k = 1,2} are the required collections.For α ∈ A and k = 1,2, define a mapping ψk

α : ClSα → C′c(ClV k

α ) by ψkα(x) = Cl(ϕ(x) ∩ V k

α ) for each x ∈ ClSα .By Lemma 5, ψk

α admits a continuous selection f kα : ClSα → Y for α ∈ A and k = 1,2. Since ψ1

α(x) ∩ ψ2α(x) = ∅,

f 1α (x) �= f 2

α (x) for each x ∈ ClSα . Thus, due to Lemma 6, ϕ admits the desired selection. �3. Selection and insertion theorems

For a space Y , let C0(Y ) be the Banach space of all continuous functions s : Y → R such that for each ε > 0 theset {y ∈ Y | |s(y)| � ε} is compact, where the linear operations are defined pointwise and ‖s‖ = sup{|s(y)| | y ∈ Y }for each s ∈ C0(Y ). If Y is the discrete space of cardinality λ, then C0(Y ) = c0(λ). For s, t ∈ C0(Y ), points sup{s, t}and inf{s, t} in C0(Y ) are defined by (sup{s, t})(y) = max{s(y), t (y)} and (inf{s, t})(y) = min{s(y), t (y)} for eachy ∈ Y , respectively (see [5, Lemma 2.7]).

Proposition 7. For every Hausdorff space Y , w(C0(Y )) � w(Y).

Proof. Let B be a base for Y such that CardB = w(Y). Let Γ be the set of pairs ({Bi}ni=1, {qi}ni=1) consisting of acollection {Bi}ni=1 of finite subsets of B and a set {qi}ni=1 of rational numbers such that Cl(

⋃B1) is compact, q1 = 0,

Cl(⋃

Bi+1) ⊂ ⋃Bi and qi < qi+1 for each i = 1,2, . . . , n − 1, n ∈ N. Note that CardΓ � CardB = w(Y) and the

collection {Bd(⋃

Bi ) | i = 1,2, . . . , n} is a finite collection of pairwise disjoint closed subsets of the compact subsetCl(

⋃B1) of Y for each ({Bi}ni=1, {qi}ni=1) ∈ Γ . For γ = ({Bi}ni=1, {qi}ni=1) ∈ Γ , we associate fγ ∈ C0(Y ) satisfying

fγ (Y \ ⋃B1) ⊂ {0}, fγ (Cl(

⋃Bi ) \ ⋃

Bi+1) ⊂ [qi, qi+1] for each i = 1,2, . . . , n − 1 and f (Cl(⋃

Bn)) ⊂ {qn} asfollows: For each i = 1,2, . . . , n − 1, define a function gi : Bd(

⋃Bi ) ∪ Bd(

⋃Bi+1) → {qi, qi+1} by gγ (y) = qk if

y ∈ Bd(⋃

Bk) and k = i, i +1. Since Cl(⋃

Bi )\⋃Bi+1 is normal, there exists a continuous extension fi : Cl(

⋃Bi )\⋃

Bi+1 → [qi, qi+1] of gi for i = 1,2, . . . , n − 1. Define fγ : Y → R by fγ (y) = 0 if y ∈ Y \ ⋃B1, fγ (y) = fi(y)

if y ∈ Cl(⋃

Bi ) \ ⋃Bi+1 and i = 1,2, . . . , n − 1, and fγ (y) = qn if y ∈ Cl(

⋃Bn). Then fγ ∈ C0(Y ) since fγ is

continuous and {y ∈ Y | fγ (y) �= 0} ⊂ Cl(⋃

B1). Thus fγ is the desired function.It suffices to show that {fγ − fγ ′ | γ, γ ′ ∈ Γ } is dense in C0(Y ). To show this, let f ∈ C0(Y ) and ε > 0. Put

f+ = sup{f,0} and f− = sup{−f,0}, where 0 : Y → R the constant function whose constant value is 0 in R. Thenf+, f− ∈ C0(Y ) and f = f+ − f−. For each i ∈ N, put Ki = {y ∈ Y | f+(y) � iε/4} and Gi = {y ∈ Y | f+(y) >

(2i − 1)ε/8}. Then Ki is compact, Gi is open in Y and Gi+1 ⊂ Ki ⊂ Gi for each i ∈ N. Since f+ is bounded,there exists n ∈ N such that Kn+1 = ∅. Take a finite subset B1 of B such that K1 ⊂ ⋃

B1 ⊂ {y ∈ Y | f+(y) � ε/8}.Then Cl(

⋃B1) is compact. For i = 2,3, . . . , n, choose a finite subset Bi of B such that Ki ⊂ ⋃

Bi ⊂ Cl(⋃

Bi ) ⊂ Gi .Then Cl(

⋃Bi+1) ⊂ ⋃

Bi for i = 1,2, . . . , n−1. Put q1 = 0 and let qi be a rational number in the open interval ((2i −1)ε/8, iε/4) for each i = 2,3, . . . , n. Then γ+ = ({Bi}ni=1, {qi}ni=1) is an element of Γ and we have that ‖f+ −fγ+‖ <

ε/2. To show this, let y ∈ Y . If y ∈ Y \ ⋃B0, then f+(y) < ε/4 and fγ+(y) = 0, and hence |f+(y) − fγ+(y)| < ε/4.

If y ∈ Cl(⋃

Bi ) \ ⋃Bi+1 for some i = 1,2, . . . , n − 1, then f+(x) ∈ ((2i − 1)ε/8, (i + 1)ε/4) since y ∈ Gi \ Ki+1,

and fγ+(y) ∈ [qi, qi+1] ⊂ ((2i − 1)ε/8, (i + 1)ε/4), and hence |f+(y) − fγ+(y)| < 3ε/8. If y ∈ Cl(⋃

Bn), thenf+(x) ∈ ((2n − 1)ε/8, (n + 1)ε/4) since y ∈ Gn and Kn+1 = ∅, and fγ+(y) = qn ∈ ((2n − 1)ε/8, nε/4), and hence|f+(y) − fγ+(y)| < 3ε/8. Similarly, we obtain γ− ∈ Γ such that ‖f− − fγ−‖ < ε/2. Then ‖f − (fγ+ − fγ−)‖ < ε.Thus {fγ − fγ ′ | γ, γ ′ ∈ Γ } is dense in C0(Y ). �

For s, t ∈ C0(Y ), we write s � t if s(y) � t (y) for each y ∈ Y , and s < t if s � t and s �= t . For spaces X

and Y , a mapping f : X → C0(Y ) is called lower (resp., upper) semicontinuous [5, Definition 2.1] if for everyx ∈ X and every ε > 0, there exists a neighborhood U of x such that if x′ ∈ U , then f (x′)(y) > f (x)(y) − ε (resp.,f (x′)(y) < f (x)(y) + ε) for each y ∈ Y .

For an infinite cardinal number λ, a Hausdorff space X is called λ-paracompact if every open cover U of X withCardU � λ is refined by a locally finite open cover of X. Theorem 4 follows from the following λ-paracompactanalogue of Theorems 1 and 2.


Theorem 8. For a T1-space X, the following statements are equivalent.

(a) X is normal and λ-paracompact.(b) For every Banach space Y with w(Y) � λ and every l.s.c. mapping ϕ : X → Fc(Y ) with Cardϕ(x) > 1 for each

x ∈ X, there exists a continuous selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X.(c) For every Hausdorff space Y with w(Y) � λ, every closed subset A of X and every two mappings g,h : A →

C0(Y ) such that g is upper semicontinuous, h is lower semicontinuous and g(x) < h(x) for each x ∈ A, thereexists a continuous mapping f : X → C0(Y ) such that g(x) < f (x) < h(x) for each x ∈ A.

(d) For every Hausdorff space Y with w(Y) � λ and every two mappings g,h : X → C0(Y ) such that g is uppersemicontinuous, h is lower semicontinuous and g(x) < h(x) for each x ∈ X, there exists a continuous mappingf : X → C0(Y ) such that g(x) < f (x) < h(x) for each x ∈ X.

Proof. To prove (a) ⇒ (b), let X be a normal λ-paracompact space, Y a Banach space with w(Y) � λ andϕ : X → Fc(Y ) an l.s.c. mapping such that Cardϕ(x) > 1 for each x ∈ X. Let D be a dense subset of Y withCardD � λ. For x ∈ X, choose εx > 0 such that diamϕ(x) > 3εx . Take y1

x , y2x ∈ D such that ‖y1

x − y2x‖ > 2εx

and ϕ(x) ∩ B(ykx , εx/4) �= ∅ for k = 1,2. Put V k

x = B(ykx , εx/4) for k = 1,2. Then we have dist(V 1

x ,V 2x ) � εx . Set

Ux = ϕ−1[V 1x ] ∩ ϕ−1[V 2

x ].Since Card{Ux | x ∈ X} � Card(D × D) � λ, we may denote {Uα | α < λ} = {Ux | x ∈ X}. For each α < λ,

choose xα ∈ X such that Uα = Uxα and let V kα = V k

xαfor k = 1,2. Take a locally finite open cover {Sα | α < λ}

of X such that ClSα ⊂ Uα for each α < λ. Consequently, we obtain a locally finite open cover {Sα | α ∈ A} of X

and a collection {V kα | α ∈ A, k = 1,2} of open convex subsets on Y such that dist(V 1

α ,V 2α ) > 0 and ϕ(x) ∩ V k

α �=∅ for each x ∈ ClSα and k = 1,2. For α ∈ A and k = 1,2, define a mapping ψk

α : ClSα → Fc(Y ) by ψkα(x) =

Cl(ϕ(x) ∩ V kα ) for each x ∈ ClSα . Then ψk

α is l.s.c. for k = 1,2. Since ClSα is normal and λ-paracompact, byvirtue of the Michael’s selection theorem [9, Theorem 3.2′′] (see also [10, Theorem 4.1]), ψk

α admits a continuousselection f k

α : ClSα → Y for k = 1,2. Since f 1α (x) �= f 2

α (x) for each x ∈ ClSα , due to Lemma 6, ϕ admits the desiredcontinuous selection.

To show (b) ⇒ (c), assume (b) and let Y , A, g and h be as in (c). Define a mapping ϕ : X → Fc(C0(Y )) byϕ(x) = [g(x),h(x)] if x ∈ A and ϕ(x) = Y otherwise. Since g(x) < h(x) for each x ∈ A, Cardϕ(x) > 1 for eachx ∈ X. By Proposition 7, w(C0(Y )) � λ. Due to [5, Lemma 2.6], ϕ is l.s.c. Thus, by (b), there exists a continuousselection f : X → C0(Y ) of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X. This f is the required mapping.

The implication (c) ⇒ (d) is obvious.To show (d) ⇒ (a), assume (d). Since C0({y}) = R, X is countably paracompact and normal due to [2, Theorem 4],

[6, Theorem 2]. H. Ohta [11] gave the positive answer to [5, Problem 5.9]. The proof in [11] also prove that condi-tion (d) implies that X is λ-paracompact. �

Theorems 2 and 8 provide the following variation of [9, Theorem 3.1′′].

Corollary 9. For a T1-space X, the following statements are equivalent.

(a) X is normal and countably paracompact.(b) For every separable Banach space Y and every l.s.c. mapping ϕ : X →Fc(Y ) with Cardϕ(x) > 1 for each x ∈ X,

there exists a continuous selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X.(c) For every separable Banach space Y and every l.s.c. mapping ϕ : X → C′

c(Y ) with Cardϕ(x) > 1 for each x ∈ X,there exists a continuous selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X.

Applying Theorem 1, V. Gutev, H. Ohta and K. Yamazaki [5, Theorem 4.6] proved that a T1-space X is perfectlynormal and λ-collectionwise normal if and only if for every generalized c0(λ)-space Y and every l.s.c. mappingϕ : X → C′

c(Y ), there exists a continuous selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X withCardϕ(x) > 1. By applying Theorem 2, instead of Theorem 1, to the proof of [5, Theorem 4.6], we have the followingcorollary.



(a) X is perfectly normal and λ-collectionwise normal.(b) For every Banach space Y with w(Y) � λ and every l.s.c. mapping ϕ : X → C′

c(Y ), there exists a continuousselection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X with Cardϕ(x) > 1.

(c) For every closed subset A of X and every two mappings g,h : X → c0(λ) such that g is upper semicontinuous, h

is lower semicontinuous and g(x) � h(x) for each x ∈ A, there exists a continuous mapping f : X → c0(λ) suchthat g(x) � f (x) � h(x) for each x ∈ A and g(x) < f (x) < h(x) whenever g(x) < h(x).

For perfectly normal λ-paracompact spaces, we have the following corollary.


(a) X is perfectly normal and λ-paracompact.(b) For every Banach space Y with w(Y) � λ and every l.s.c. mapping ϕ : X → Fc(Y ), there exists a continuous

selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X with Cardϕ(x) > 1.(c) For every Hausdorff space Y with w(Y) � λ, every closed subset A of X and every two mappings g,h : A →

C0(Y ) such that g is upper semicontinuous, h is lower semicontinuous and g(x) � h(x) for each x ∈ A, thereexists a continuous mapping f : X → C0(Y ) such that g(x) � f (x) � h(x) for each x ∈ A and g(x) < f (x) <

h(x) whenever g(x) < h(x).(d) For every Hausdorff space Y with w(Y) � λ and every two mappings g,h : X → C0(Y ) such that g is upper

semicontinuous, h is lower semicontinuous and g(x) � h(x) for each x ∈ A, there exists a continuous mappingf : X → C0(Y ) such that g(x) � f (x) � h(x) for each x ∈ A and g(x) < f (x) < h(x) whenever g(x) < h(x).

Proof. The implication (a) ⇒ (b) can be proved by applying Theorem 8 to the proof of (1) ⇒ (2) of [5, Theorem 4.6].Proof of (b) ⇒ (c) is similar to one of (b) ⇒ (c) in Theorem 8. The implication (c) ⇒ (d) is obvious. To see (d) ⇒(a), assume (d). Since C0({y}) = R, X is perfectly normal due to [9, Theorem 3.1′′′]. By Theorem 8, X is λ-para-compact. �

Finally, concerning [9, Proposition 1.4], we consider the following problem on the selection extension propertyavoiding extreme points.

Problem 12. Let X be a countably paracompact λ-collectionwise normal space (resp., normal λ-paracompact space),A a closed subset of X, Y a Banach space with w(Y) � λ and ϕ : X → C′

c(Y ) (resp., ϕ : X → Fc(Y )) an l.s.c. mappingwith Cardϕ(x) > 1 for each x ∈ X. Is every continuous selection f : A → Y of ϕ|A with f (x) ∈ wci(ϕ(x)) for eachx ∈ A extended to a continuous selection f : X → Y of ϕ such that f (x) ∈ wci(ϕ(x)) for each x ∈ X?

If, in addition, X is perfectly normal, then Problem 12 is affirmative by Corollaries 10 and 11, respectively. On theother hand, the following sandwich-like properties are valid.

Proposition 13. If X is a countably paracompact λ-collectionwise normal space, A and B are closed subsets of X andg,h : B → c0(λ) are two mappings such that g is upper semicontinuous, h is lower semicontinuous and g(x) < h(x)

for each x ∈ B , then every continuous mapping f : A → c0(λ) with g(x) < f (x) < h(x) for each x ∈ A ∩ B isextended to a continuous mapping f : X → c0(λ) such that g(x) < f (x) < h(x) for each x ∈ B .

Proposition 14. If X is a normal λ-paracompact space, A and B are closed subsets of X, Y is a Hausdorff space withw(Y) � λ and g,h : B → C0(Y ) are two mappings such that g is upper semicontinuous, h is lower semicontinuousand g(x) < h(x) for each x ∈ B , then every continuous mapping f : A → c0(λ) with g(x) < f (x) < h(x) for eachx ∈ A ∩ B is extended to a continuous mapping f : X → C0(Y ) such that g(x) < f (x) < h(x) for each x ∈ B .

Here, we prove Proposition 14. Proposition 13 can be proved similarly.


Proof of Proposition 14. Let X, A, B , Y , g, h and f be as in Proposition 14. Define mappings g, h : B → C0(Y )

by g(x) = f (x) if x ∈ A ∩ B and g(x) = g(x) otherwise, and h(x) = f (x) if x ∈ A ∩ B and h(x) = h(x) other-wise, respectively. Then g is upper semicontinuous, h is lower semicontinuous and g(x) < h(x) and g(x) < h(x) foreach x ∈ B . By Theorem 8, there exist continuous functions r1, r2 : X → C0(Y ) such that g(x) < r2(x) < h(x) andg(x) < r1(x) < h(x) for each x ∈ B . Note that r1(x) < f (x) < r2(x) for each x ∈ A ∩ B . Due to [12, Proposition 1,Corollary 2], there exists a continuous extension f1 : B → C0(Y ) of f |A∩B . Define a mapping f2 : B → C0(Y ) byf2(x) = inf{sup{r1(x), f1(x)}, r2(x)} for each x ∈ X. Due to [5, Lemma 2.8], f2 is a continuous extension of f |A∩B .Let f3 : A∪B → C0(Y ) be the mapping defined by f3(x) = f2(x) if x ∈ B and f3(x) = f (x) otherwise. Since both A

and B are closed subsets of X, f3 is a continuous extension of f . Finally, by applying [12, Proposition 1, Corollary 2],we have a continuous extension f : X → C0(Y ) of f3. This f is the desired mapping. �

The author does not know whether Problem 12 is affirmative in general.

Acknowledgements

The author would like to thank Professor Haruto Ohta for his suggestions, which improve the proof of Theorem 2.The author also wish to thank Professors Valentin Gutev, Takao Hoshina and Kaori Yamazaki and the referee for theirvaluable comments.

References

[1] M. Coban, V. Valov, A certain theorem of Michael on sections, C. R. Acad. Bulgare Sci. 28 (1975) 871–873 (in Russian).[2] C.H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951) 219–224.[3] C.H. Dowker, On a theorem of Hanner, Ark. Mat. 2 (1952) 307–313.[4] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.[5] V. Gutev, H. Ohta, K. Yamazaki, Selections and sandwich-like properties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55

(2003) 499–521.[6] M. Katetov, On real-valued functions in topological spaces, Fund. Math. 38 (1951) 85–91.[7] R.E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, vol. 183, Springer, New York, 1998.[8] E. Michael, Point-finite and locally finite coverings, Canad. J. Math. 7 (1955) 275–279.[9] E. Michael, Continuous selections I, Ann. of Math. 63 (1956) 361–382.

[10] S. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980) 291–317.[11] H. Ohta, An insertion theorem characterizing paracompactness, Topology Proc. 30 (2006) 557–564.[12] T. Przymusinski, Collectionwise normality and extensions of continuous functions, Fund. Math. 98 (1978) 75–81.

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