elementary topology - american mathematical society · the subject of the book: elementary topology...

Elementary Topolog y Problem Textbook

This page intentionally left blank

Elementary Topolog y Problem Textbook

O. Ya. Viro O. A. Ivanov N. Yu. Netsvetaev V. M. Kharlamov

>AMS AMERICAN MATHEMATICA L SOCIET Y

Providence, Rhode Island

http://dx.doi.org/10.1090/mbk/054

2000 Mathematics Subject Classification. P r i m a r y 54 -01 , 54-00 , 55-00 , 55 -01 , 57 -01 , 57M05, 57M10 , 57M15 .

T h e cove r desig n i s base d o n a sketc h b y Mash a Netsvetaev a an d Niki t a Netsvetaev . T h e pho t o o n p . xvi i o f Vladimi r Abramovic h Rokhli n i s coutes y o f Ole g Viro . T h e pho t o o n th e bac k cove r an d o n p . xx o f Ole g Yanovic h Viro , Viatchesla v Mikhailovic h

Khar lamov, Nikit a Yur'evic h Netsvetaev , an d Ole g Aleksandrovic h Ivano v i s courtes y o f Jul ia Viro .

For addit iona l informatio n an d upda t e s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / m b k - 5 4

Library o f Congres s Cataloging-in-Publicatio n D a t a

Elementary topolog y : proble m textboo k / O . Ya . Viro . . . [et al.] . p. cm .

Includes bibliographica l reference s an d index . ISBN 978-0-8218-4506- 6 (alk . paper ) 1. Topology—Textbooks . I . Viro , O . IA. , 1948 -

QA611.E534 200 8 514—dc22 200800930 3

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] .

© 200 8 b y th e authors . Al l right s reserved . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durabilit y

Visit th e AM S hom e pag e a t ht tp: / /www.ams.org /

10 9 8 7 6 5 4 3 2 1 1 3 1 2 1 1 1 0 0 9 0 8

Dedicated to the memory of Vladimir Abramovich Rokhlin (1919-1984) - our teacher

Contents

Introduction

Part 1 . Genera l Topolog y

Chapter I . Structure s an d Space s

1. Set-Theoreti c Digression : Set s

2. Topolog y o n a Se t

3. Base s

4. Metri c Space s

5. Subspace s

6. Positio n o f a Poin t wit h Respec t t o a

7. Ordere d Set s

8. Cycli c Order s

Proofs an d Comment s

Chapter II . Continuit y

9. Set-Theoreti c Digression : Map s

10. Continuou s Map s

11. Homeomorphism s

Proofs an d Comment s

Chapter III . Topologica l Propertie s

12. Connectednes s

13. Applicatio n o f Connectednes s

V l l l Contents

14. Pat h Connectednes s 9 2

15. Separatio n Axiom s 9 7

16. Countabilit y Axiom s 10 3

17. Compactnes s 10 8

18. Sequentia l Compactnes s 11 4

19x. Loca l Compactnes s an d Paracompactnes s 11 7

Proofs an d Comment s 12 2

Chapter IV . Topologica l Construction s 13 5

20. Multiplicatio n 13 5

21. Quotien t Space s 14 1

22. Zo o of Quotien t Space s 14 5

23. Projectiv e Space s 15 5

24x. Finit e Topologica l Space s 15 9

25x. Space s o f Continuou s Map s 16 3


Chapter V . Topologica l Algebr a 17 9

26x. Generalitie s o n Group s 18 1

27x. Topologica l Group s 18 7

28x. Construction s 19 1

29x. Action s o f Topologica l Group s 19 6


Part 2 . Element s o f Algebrai c Topolog y

Chapter VI . Fundamenta l Grou p 20 7

30. Homotop y 20 7

31. Homotop y Propertie s o f Path Multiplicatio n 21 2

32. Fundamenta l Grou p 21 5

33. Th e Rol e o f Bas e Poin t 22 0


Chapter VII . Coverin g Spaces and Calculatio n o f Fundamental Group s 231

34. Coverin g Space s 23 1

35. Theorem s o n Pat h Liftin g 23 5

36. Calculatio n o f Fundamenta l Group s b y Usin g Universa l Coverings 23 7

Contents IX


Chapter VIII . Fundamenta l Grou p an d Map s 24 7

37. Induce d Homomorphism s

and Thei r Firs t Application s 24 7

38. Retraction s an d Fixe d Point s 25 3

39. Homotop y Equivalence s 25 6

40. Coverin g Space s vi a Fundamenta l Group s 26 1

41x. Classificatio n o f Coverin g Space s 26 3


Chapter IX . Cellula r Technique s 27 9

42. Cellula r Space s 27 9

43x. Topologica l Propertie s o f Cellula r Space s 28 6

44. Cellula r Construction s 28 8

45. One-Dimensiona l Cellula r Space s 29 1

46. Fundamenta l Grou p o f a Cellula r Spac e 29 5


Hints, Comments , Advices , Solutions , an d Answer s 31 7

Bibliography 39 3

Index 39 5

Introduction

The subjec t o f th e book : Elementar y Topolog y

Elementary mean s clos e t o elements , basics . I t i s impossibl e t o deter -mine precisely , onc e an d fo r all , whic h topolog y i s elementar y an d whic h is not . Th e elementar y par t o f a subjec t i s th e par t wit h whic h a n exper t starts t o teac h a novice .

We suppose tha t ou r studen t i s ready t o stud y topology . So , we do no t try t o win her o r his attention an d benevolenc e by hasty an d obscur e storie s about mysteriou s an d attractiv e thing s suc h a s th e Klei n bottle, 1 thoug h the Klei n bottl e wil l appea r i n it s turn . However , w e star t wit h wha t a topological spac e is , that is , we star t wit h genera l topology .

General topolog y becam e a par t o f th e genera l mathematica l languag e a lon g tim e ago . I t teache s on e t o spea k clearl y an d precisel y abou t thing s related t o th e ide a o f continuity . I t i s no t onl y neede d t o explai n what , finally, th e Klei n bottl e is , bu t i t i s als o a wa y t o introduc e geometrica l images int o an y are a o f mathematics , n o matte r ho w far fro m geometr y th e area ma y b e a t first glance .

As a n activ e researc h area , genera l topolog y i s practicall y completed . A permanent usag e i n the capacit y o f a genera l mathematica l languag e ha s polished it s system of definitions an d theorems. Indeed , nowadays , the stud y of genera l topolog y resemble s a stud y o f a languag e rathe r tha n a stud y o f mathematics: on e ha s t o lear n man y ne w words , whil e th e proof s o f th e majority o f th e theorem s ar e extremel y simple . However , th e quantit y o f

1A perso n wh o i s lookin g fo r suc h elementar y topolog y wil l easil y find i t i n numerou s book s with beautifu l picture s o n visua l topology .

XI

X l l Introduction

the theorem s i s huge. Thi s come s a s n o surpris e becaus e the y pla y th e rol e of rules tha t regulat e usag e o f words .

The boo k consist s o f two parts . Genera l topolog y i s the subjec t o f par t one. Th e secon d par t i s a n introductio n t o algebrai c topolog y vi a it s mos t classical and elementar y segment , whic h emerges fro m th e notion s o f funda -mental grou p an d coverin g space .

In our opinion, elementary topology als o includes basic topology of man-ifolds, i.e. , space s tha t loo k locall y a s th e Euclidea n space . One - an d two -dimensional manifolds , i.e. , curve s an d surface s ar e especiall y elementary . However, a boo k shoul d no t b e to o thick , an d s o we had t o stop .

Chapter 5 , whic h i s th e las t chapte r o f th e firs t part , keep s somewha t aloof. I t i s devote d t o topologica l groups . Th e materia l i s intimatel y re -lated t o a numbe r o f differen t area s o f Mathematics . Althoug h topologica l groups pla y a profoun d rol e i n thos e areas , i t i s no t tha t importan t i n th e initial stud y o f genera l topology . Therefore , masterin g thi s materia l ma y be postpone d unti l i t appear s i n a substantia l wa y i n othe r mathematica l courses (whic h wil l concer n th e Li e groups , functiona l analysis , etc.) . Th e main reason why we included thi s materia l i s that i t provide s a great variet y of examples an d exercises .

Organization o f th e tex t

Even a cursor y overvie w detect s unusua l feature s i n the organizatio n o f this book . W e dared t o com e up with severa l innovations an d hop e tha t th e reader wil l quickl y ge t use d t o the m an d eve n fin d the m useful .

We kno w tha t th e need s an d interest s o f ou r reader s vary , an d realiz e that i t i s very difficul t t o make a book interesting and usefu l fo r each reader . To solve this problem , w e formatted th e tex t i n suc h a way tha t th e reade r could easil y determin e wha t (s)h e ca n expec t fro m eac h piec e o f th e text . We hope tha t thi s wil l allow the reade r t o organiz e studyin g the materia l of the boo k i n accordanc e wit h hi s o r he r taste s an d abilities . T o achiev e thi s goal, w e use severa l tricks .

First o f all , w e distinguishe d th e basic , s o t o speak , lectur e line . Thi s is th e materia l whic h w e conside r basic . I t constitute s a mino r par t o f th e text.

The basi c materia l i s often interrupte d b y specific examples , illustrativ e and trainin g problems , an d discussio n o f th e notion s tha t ar e relate d t o these example s an d problems , bu t ar e no t use d i n wha t follows . Som e o f the notion s pla y a fundamental rol e in other area s o f mathematics, bu t her e they ar e o f minor importance .

xm

In a word , th e basi c lin e i s interrupte d b y variations whereve r possi -ble. Th e variation s ar e clearl y separate d fro m th e basic theme b y graphica l means.

The second feature distinguishin g the presen t boo k from th e majorit y o f other textbook s i s that proof s ar e separate d fro m formulations . Thi s make s the boo k loo k like a pure problem book . I t woul d b e easy to mak e the boo k looking lik e hundred s o f othe r mathematica l textbooks . Fo r thi s purpose , it suffice s t o mov e al l variation s t o th e end s o f thei r section s s o tha t the y would loo k lik e exercise s t o th e basi c text , an d pu t th e proof s o f theorem s immediately afte r thei r formulations .

For w h o m i s thi s book ?

A reader who has safely reache d the university leve l in her/his educatio n may bravel y approac h thi s book . Supe r brav e daredevil s ma y tr y i t eve n earlier. However , w e cannot sa y tha t n o preliminar y knowledg e i s required . We suppose tha t th e reade r i s familiar wit h rea l numbers , and , surely , wit h natural, integer , an d rationa l number s too . A knowledg e o f comple x num -bers woul d als o b e useful , althoug h on e ca n manag e withou t the m i n th e first par t o f the book .

We assume that th e reader is acquainted with naive set theory, but admi t that thi s acquaintanc e ma y b e superficial . Fo r thi s reason , w e make specia l set-theoretical digression s wher e th e knowledg e o f se t theor y i s particularl y desirable.

We d o no t seriousl y rel y o n calculus , bu t becaus e th e majorit y o f ou r readers ar e alread y familia r wit h it , a t leas t slightly , w e d o no t hesitat e t o resort t o usin g notation s an d notion s fro m calculus .

In th e secon d part , experienc e i n grou p theor y wil l b e useful , althoug h we give al l necessary informatio n abou t groups .

One of the mos t valuabl e acquisition s tha t th e reade r ca n make by mas-tering th e presen t boo k i s ne w element s o f mathematica l cultur e an d a n ability t o understan d an d appreciat e a n abstrac t axiomati c theory . Th e higher th e degre e i n whic h th e reade r alread y possesse s thi s ability , th e easier i t wil l be fo r he r o r hi m t o maste r th e materia l o f the book .

If yo u wan t t o stud y topolog y o n you r own , d o tr y t o wor k wit h th e book. I t ma y tur n ou t t o b e precisel y wha t yo u need . However , yo u shoul d attentively rerea d th e res t o f the Introductio n agai n i n orde r t o understan d how th e materia l i s organized an d ho w yo u ca n us e it .

XIV Introduction

The basi c them e

The cor e o f the boo k i s made u p o f the materia l o f the topolog y cours e for student s majorin g i n Mathematic s a t th e Sain t Petersbur g (Leningrad ) State University . Th e cor e materia l make s u p a relativel y smal l par t o f th e book an d involve s nearl y n o complicate d arguments .

The reader should not think that b y selecting the basic theme the author s just tr y t o impos e thei r taste s o n he r o r him . W e do not hesitat e t o d o thi s occasionally, bu t her e ou r primar y goa l i s to organiz e stud y o f the subject .

The basi c theme form s a complete entity . Th e reade r wh o has mastere d the basi c them e ha s mastere d th e subject . Whethe r th e reade r ha d looke d in th e variation s o r no t i s her o r hi s business . However , th e variation s hav e been include d i n order t o hel p the reade r wit h masterin g the basi c material . They ar e no t exile d t o th e final page s o f section s i n orde r t o hav e the m a t hand precisel y whe n the y ar e mos t needed . B y th e way , th e variation s ca n tell yo u abou t man y interestin g things . However , followin g th e variation s too literall y an d carefull y ma y tak e fa r to o long .

We believe that th e materia l presented i n the basic theme is the minima l amount o f topology that mus t b e mastered by every student wh o has decided to becom e a professiona l mathematician .

Certainly, a student whos e interests will be related to topology and othe r geometrical discipline s wil l hav e t o lear n fa r mor e tha n th e basi c them e includes. I n thi s cas e th e materia l ca n serv e a s a goo d startin g point .

For a student wh o is not goin g to becom e a professional mathematician , even a selective acquaintanc e wit h th e basi c them e migh t b e useful . I t ma y be usefu l fo r preparatio n fo r a n exa m o r jus t fo r catchin g a glimps e an d a feeling o f abstract mathematics , with it s emphasized valu e of definitions an d precise formulations .

W h e r e ar e th e proofs ?

The boo k i s tailored fo r a reade r wh o i s determined t o wor k actively .

The proofs of theorems are separated from their formulations and placed at the end of the current chapter.

We believ e tha t th e firs t reactio n t o th e formulatio n o f an y assertio n (coming immediatel y afte r th e feelin g tha t th e formulatio n ha s bee n under -stood) mus t b e a n attemp t t o prov e th e assertion—o r t o disprov e it , i f yo u do no t manag e t o prov e it . A n attemp t t o disprov e a n assertio n ma y b e useful bot h fo r achievin g a bette r understandin g o f the formulatio n an d fo r looking fo r a proof .

By keepin g th e proof s awa y fro m th e formulations , w e want t o encour -age th e reade r t o thin k throug h eac h formulation , and , o n th e othe r hand ,

XV

to mak e th e boo k inconvenien t fo r careles s skimming . However , a reade r who prefer s a mor e traditiona l styl e and , fo r som e reason , doe s no t wis h t o work to o activel y ca n eithe r find th e proof s a t th e en d o f th e chapter , o r skip them al l together. (Certainly , i n the latte r cas e there i s some danger of misunderstanding.)

This styl e ca n als o pleas e a n exper t wh o need s a handboo k an d prefer s formulations no t overshadowe d b y proofs. Mos t o f the proofs ar e simple an d easy t o discover .

Structure o f th e boo k

Basic structura l unit s o f th e boo k ar e sections . The y ar e divide d int o numbered an d title d subsections . Eac h subsectio n i s devote d t o a singl e topic an d consist s o f definitions , comments , theorems , exercises , problems , and riddles .

By a riddle we mean a problem whose solution (an d ofte n als o the mean -ing) shoul d b e guesse d rathe r tha n calculate d o r deduce d fro m th e formula -tion.

Theorems, exercises , problems , an d riddle s belongin g to th e basi c mate -rial ar e numbere d b y pair s consistin g o f th e numbe r o f th e curren t sectio n and a letter , separate d b y a dot .

2.B. Riddle. Takin g int o accoun t th e numbe r o f th e riddle , determin e i n which sectio n i t mus t b e contained . B y th e way , i s this reall y a riddle ?

The letter s ar e assigne d i n alphabetica l order . The y numbe r th e assertion s inside a section .

A difficul t proble m (o r theorem ) i s ofte n followe d b y a sequenc e o f as -sertions tha t ar e lemma s t o th e problem . Suc h a chai n ofte n end s wit h a problem in which we suggest the reader, arme d with the lemmas just proven , return t o th e initia l proble m (respectively , theorem) .

Variations

The basi c materia l i s surrounde d b y numerou s trainin g problem s an d additional definitions , theorems , an d assertions . I n spit e o f thei r relatio n to th e basi c material , the y usuall y ar e lef t outsid e o f th e standar d lectur e course.

Such additiona l materia l i s easy t o recogniz e i n the boo k b y th e smalle r prin t and wid e margins , a s show n here . Exercises , problems , an d riddle s tha t ar e no t included i n th e basi c material , bu t ar e closel y relate d t o it , ar e numbere d b y pair s consisting o f the numbe r o f a section an d th e numbe r o f the assertio n i n the limit s of th e section .

2.5. Fin d a proble m wit h th e sam e numbe r 2.5 i n th e mai n bod y o f th e book .

XVI Introduction

All solutions to problems are located at the end of the book.

As i s common , th e problem s tha t hav e seeme d t o b e mos t difficul t t o the author s ar e marke d b y a n asterisk . The y ar e include d wit h differen t purposes: t o outlin e relation s t o othe r area s o f mathematics , t o indicat e possible direction s o f development o f the subject , o r just t o pleas e a n ambi -tious reader .

Additional theme s

We decided to make accessible for intereste d student s certai n theoretica l topics complementin g th e basi c material . I t woul d b e natura l t o includ e them int o lectur e course s designe d fo r senio r (o r graduate ) students . How -ever, thi s doe s no t usuall y happen , becaus e th e topic s d o no t fit wel l int o traditional graduat e courses . Furthermore , studyin g them seem s to be mor e natural durin g th e ver y first contact s wit h topology .

In the book, such topics are separated int o individual subsections , whose numbers contai n th e symbo l x , whic h mean s extra. (Sometimes , a whol e section i s marked i n this way , and , i n one case , eve n a whole chapter. )

Certainly, regardin g thi s materia l a s additiona l i s a matte r o f taste an d viewpoint. Qualifyin g a topi c a s additional , w e follow ou r ow n idea s abou t what mus t b e containe d i n th e initia l stud y o f topology . W e realiz e tha t some (i f no t most ) o f ou r colleague s ma y disagre e wit h ou r choice , bu t w e hope tha t ou r decoration s wil l no t hinde r the m fro m usin g the book .

Advices t o th e reade r

You ca n us e th e presen t boo k whe n preparin g fo r a n exa m i n topolog y (especially s o i f th e exa m consist s i n solvin g problems) . However , i f yo u attend lecture s i n topology , the n i t i s reasonabl e t o rea d th e boo k befor e the lectures , an d tr y t o prov e th e assertion s i n i t o n you r ow n befor e th e lecturer wil l prove them .

The reade r wh o ca n prov e assertion s o f th e basi c them e o n hi s o r he r own needn' t solv e all o f th e problem s suggeste d i n th e variations , an d ca n resort t o a brie f acquaintanc e wit h thei r formulation s an d solv e onl y th e most difficul t o f them . O n th e othe r hand , th e mor e difficul t i t i s fo r yo u to prov e assertion s o f th e basi c theme , th e mor e attentio n yo u shoul d pa y to illustrativ e problems , an d th e les s attentio n shoul d b e pai d t o problem s with a n asterisk .

Many o f ou r illustrativ e problem s ar e eas y t o com e u p with . Moreover , when seriousl y studyin g a subject , on e shoul d permanentl y coo k u p ques -tions o f this kind .

XVII

On th e othe r hand , som e problem s presente d i n th e boo k ar e no t eas y to com e u p wit h a t all . W e have widel y use d al l kind s o f sources , includin g both literatur e an d teachers ' folklore .

Notations

We di d ou r bes t t o avoi d notation s whic h ar e no t commonl y accepted . The onl y exceptio n i s th e us e o f a fe w symbol s whic h ar e ver y convenien t and almos t self-explanatory . Namely , withi n proof s symbol s (=» ) an d (<=) shoul d b e understoo d a s (sub)titles . Eac h o f the m mean s tha t w e start provin g the correspondin g implication . Similarly , symbol s | C | and | p | indicate th e beginnin g o f proofs o f the correspondin g inclusions .

How thi s boo k wa s create d

In the basic theme, we follow the course of lectures composed by Vladimir Abramovich Rokhli n a t th e Facult y o f Mathematic s an d Mechanic s o f th e Leningrad Stat e Universit y i n the 1960s . I t seem s appropriat e t o sketc h th e circumstances o f creating the course , althoug h we started t o write this boo k only afte r Vladimi r Abramovich' s deat h (1984) .

Vladimir Abramovic h Rokhli n give s a lecture , 1960s .

XV111 Introduction

In the 1960s , mathematics was one of the most attractiv e area s of science for youn g peopl e i n th e Sovie t Union , bein g secon d mayb e onl y t o physic s among the natura l sciences . Ever y year mor e than a hundred student s wer e enrolled i n th e mathematica l subdivisio n o f the Faculty .

Several dozen of them were alumnae and alumni of mathematical schools. The system an d content s o f the lecture courses a t th e Faculty were seriousl y updated.

Until Rokhli n develope d hi s course , topolog y wa s taugh t i n the Facult y only i n th e framewor k o f specia l courses . Rokhli n succeede d i n includin g a one-semeste r cours e o n topolog y int o th e syste m o f genera l mandator y courses. Th e cours e consiste d o f thre e chapter s devote d t o genera l topol -ogy, fundamenta l grou p an d coverings , an d manifolds , respectively . Th e contents o f th e firs t tw o chapter s differe d onl y slightl y fro m th e basi c ma -terial o f th e book . Th e las t chapte r starte d wit h a genera l definitio n o f a topological manifold , include d a topological classification o f one-dimensiona l manifolds, an d ende d eithe r wit h a topological classificatio n o f triangulate d two-dimensional manifold s o r wit h element s o f differentia l topology , u p t o embedding a smooth manifol d i n the Euclidea n space .

Three o f the fou r author s belon g to th e firs t generatio n o f students wh o attended Rokhlin' s lectur e course . Thi s wa s a one-semeste r course , thre e hours a week in the firs t semeste r o f the secon d year . A t mos t tw o two-hou r lessons during th e whol e semeste r wer e devoted t o solvin g problems . I t wa s not Rokhlin , bu t hi s graduat e student s wh o conducte d thes e lessons . Fo r instance, i n 1966-6 8 they wer e conducted b y Mish a Gromov—a n outstand -ing geometer, currentl y a professor o f the Pari s Institute de s Hautes Etudie s Scientifiques an d th e Ne w Yor k Couran t Institute . Rokhli n regarde d th e course a s a theoretica l on e an d di d no t wis h t o spen d lectur e tim e solvin g problems. Indeed , i n the framewor k o f the cours e on e di d no t hav e to teac h students ho w to solve series of routine problems , like problems in technique s of differentiatio n an d integration , tha t ar e traditiona l fo r calculus .

Despite th e fac t tha t w e buil t ou r boo k b y startin g fro m Rokhlin' s lec -tures, th e boo k wil l giv e yo u n o ide a abou t Rokhlin' s style . Th e lecture s were brilliant . Rokhli n wrot e ver y littl e o n the blackboard . Nevertheless , i t was very easy to take notes. H e spoke without haste , with maximall y simpl e and ideall y correc t sentences .

For th e las t time , Rokhli n gav e hi s mandatory topolog y cours e i n 1973 . In Augus t o f 1974 , becaus e o f hi s seriou s illness , th e administratio n o f th e Faculty ha d t o loo k fo r a perso n wh o woul d substitut e fo r Rokhli n a s a lecturer. Th e proble m wa s complicate d b y th e fac t tha t th e result s o f th e exams i n the precedin g yea r wer e terrible. I n 1973 , the tim e allotte d fo r th e course was increased u p t o fou r hour s a week, whil e the numbe r o f student s

XIX

had grown , and , respectively , th e leve l o f thei r trainin g ha d decreased . A s a result , th e grade s fo r exam s "crashe d down" .

It wa s decide d tha t th e whol e class , whic h consiste d o f abou t 17 5 stu -dents, shoul d b e spli t int o tw o classes . Professo r Vikto r Zalgalle r wa s ap -pointed t o giv e lecture s t o th e student s wh o wer e goin g t o specializ e i n applied mathematics , whil e Assistan t Professo r Ole g Vir o woul d giv e th e lectures t o student-mathematicians . Zalgalle r suggeste d introducin g exer -cise lessons—one hou r a week. A s a result , th e tim e allotted fo r th e lecture s decreased, an d d e fact o th e volum e o f the materia l als o reduced alon g wit h the time .

It remaine d to understand what to do in the exercise lessons. On e had t o develop a system o f problems an d exercise s tha t woul d giv e an opportunit y to revisit the definitions give n in the lectures, and would allow one to develo p skills in proving eas y theorems fro m genera l topolog y i n the framewor k o f a simple axiomati c theory .

Problems i n th e first par t o f the boo k ar e a resul t o f ou r effort s i n thi s direction. Gradually , exercis e lesson s an d problem s wer e becomin g mor e and mor e usefu l a s lon g a s w e ha d t o teac h student s wit h a lowe r leve l o f preliminary training . I n 1988 , the Publishin g Hous e o f the Leningra d Stat e University publishe d th e problem s i n a smal l book , Problems in Topology.

Students foun d th e boo k useful . On e o f them , Alekse i Solov'ev , eve n translated i t int o Englis h o n hi s ow n initiativ e whe n h e becam e a gradu -ate studen t a t th e Universit y o f California . Th e translatio n initiate d a ne w stage o f work o n th e book . W e started developin g th e Russia n an d Englis h versions i n paralle l an d practicall y covere d th e entir e materia l o f Rokhlin' s course. I n 2000 , th e Publishin g Hous e o f th e Sain t Petersbur g Stat e Uni -versity publishe d th e secon d Russia n editio n o f th e book , whic h alread y included a chapte r o n the fundamenta l grou p an d coverings .

The Englis h versio n wa s use d b y Ole g Vir o fo r hi s lectur e cours e i n th e USA (Universit y o f California) an d Swede n (Uppsal a University) . Th e Rus -sian versio n wa s use d b y Slav a Kharlamo v fo r hi s lectur e course s i n Franc e (Strasbourg University) . Th e lecture s hav e bee n give n fo r quit e differen t audiences: bot h fo r undergraduat e an d graduat e students . Furthermore , few professor s (som e of whom th e author s hav e no t know n personally ) hav e asked th e authors ' permissio n t o us e th e Englis h versio n i n thei r lectures , both i n th e countrie s mentione d abov e an d i n othe r ones . Ne w demand s upon th e tex t hav e arisen . Fo r instance , w e were aske d t o includ e solution s to problem s an d proof s o f theorem s i n th e book , i n orde r t o mak e i t mee t the Wester n standard s an d transfor m i t fro m a proble m boo k int o a self -sufficient textbook . Afte r som e hesitation , w e fulfille d thos e requests , th e

X X Introduction

more s o tha t the y wer e uphel d b y th e Publishin g Hous e o f th e America n Mathematical Society .

Acknowledgments

We are grateful t o all of our colleagues for their advices and help. Mikhai l Zvagel'skii, Anatoli ! Korchagin , Seme n Podkorytov , an d Alexande r Shu -makovitch mad e numerou s usefu l remark s an d suggestions . W e als o than k Aleksei Solov'e v fo r translatin g th e firs t editio n o f th e boo k int o English . Our specia l gratitud e i s du e t o Vikto r Abramovic h Zalgaller , whos e peda -gogical experienc e an d sincer e wis h t o hel p playe d a n invaluabl e rol e fo r u s at a time whe n w e were young .

Each o f u s ha s bee n luck y t o b e a studen t o f Vladimi r Abramovic h Rokhlin, t o whos e memor y w e dedicate thi s book .

The authors , fro m th e lef t t o th e right : Oleg Yanovich Viro ,

Viatcheslav Mikhailovic h Kharlamov , Nikita Yur'evic h Netsvetaev , Oleg Aleksandrovic h Ivanov .

Bibliography

[1] A . V. Arkhangel'skii , V . I . Ponomarev , Fundamentals of General Topology: Problems and Exercises, Kluwe r Academi c Print , 2001 .

[2] D . B . Fuks , V . A . Rokhlin , Beginner's Course in Topology, Geometric Chapters, Springer-Ver lag, 1984 .

[3] R . L . Graham, B . L. Rotschild, an d J . H . Spencer, Ramsey Theory, Joh n Wiley , 1990 .

[4] A . G . Kurosh , The Theory of Groups, Vol . 1-2 , Chelsea , Ne w York , 1956 , 1960 .

[5] W . S . Massey , Algebraic Topology: An Introduction, Hartcourt , Brac e & World, Inc. , 1967.

[6] J . R . Munkres , Topology, Prentic e Hall , Inc. , 2000 .

[7] L . A . Steen , J . A . Seebach , Jr. , Counterexamples in Topology, Springer-Ve r lag, 1978 .

[8] O . Ya . Viro , O . A . Ivanov , N . Yu. Netsvetaev , V . M . Kharlamov , Problems in Topol-ogy, Leningra d Stat e Universit y Press , 1988 . - 9 2 p . [I n Russian ]

393

Index

0 5

tyl4

B r (a ) 1 9 C 5 CPn 15 6 CI 30 Dn 2 0 Fr 3 0 H 15 7 H P n 15 8 Int 2 9 N 5 Q 5 R 5 Rn 1 8 MP n 15 5 R T l 1 4 5 n 2 0 Z 5 abbreviation o f a ma p 5 8 accumulation poin t 11 4 action

properly discontinuou s 19 7 group i n a se t 19 6

transitive 19 6 effective 19 6 faithful 19 6

continuous 19 7 of a fundamenta l grou p i n a fiber 26 5

adherent poin t 3 0 Alexandrov compactificatio n 11 8 arrow 1 1 asymmetric 2 5 attaching ma p 151 , 281 attaching o f a spac e 15 1 automorphism o f coverin g 26 5

axioms o f coun t ability 10 3 first 10 5 second 10 4 separability 10 4

axioms o f grou p 18 1 axioms o f topologica l structur e 1 1 ball

open 1 9 closed 1 9

baricentric subdivisio n 16 2 of a pose t 16 2

base fo r a topologica l structur e 1 6 base a t a poin t 10 5 base o f a coverin g 23 1 basic surface s 153 , 28 4 bicompactness 10 8 bijection 5 5 Borsuk Theore m 25 4 Borsuk-Ulam Theore m 25 2 bouquet 255 , 23 8 boundary

of a se t 30 , 8 6 point o f a se t 2 9

bounded se t 2 1 Brower Theore m 25 5 Cantor se t 1 5 cardinality 10 3 Cartesian produc t 135 , se e produc t Cauchy sequenc e 11 5 cell 27 9

closed 27 9 dividing 29 2

n-cell 27 9 cellular

decomposition 28 1 space 27 9

396 Index

O-dimensional 27 9 1-dimensional 29 1

subspace 28 1 center o f a grou p 19 2 circular loo p 21 6 closed

set 1 3 in a subspac e 2 7

map 11 2 closure

of a se t 3 0 in a metri c spac e 3 0 operation o f 31 , 32 sequential o f a se t 10 6

coincidence se t o f map s 9 8 collapse 28 8 compact

cellular spac e 28 7 set 10 9 space 10 8

compactification 11 8 Alexandrov 11 8 one-point 11 8

complement o f a se t 1 0 complete metri c spac e 11 5 complex projectiv e lin e 15 6 component 8 5

connected 8 5 path connecte d 9 4

composition o f map s 5 7 cone 3 7 conjugation 18 8 connected

cellular spac e 28 7 component 8 5 set 8 4 space 8 3

contraction o f a se t 145 , 14 6 coset 18 5 count ability axio m 10 3

the firs t 10 5 the secon d 10 4 in a topologica l grou p 18 9

cover 6 3 closed 6 4 dominates a partio n o f unit y 12 0 fundamental 63 , 28 6 inscribed 64 , 11 9 locally finit e 64 , 11 9 open 6 4 refinement o f othe r on e 6 4

covering 231 , 6 3 base o f 23 1 ra-fold 23 4 induced 26 7 in narro w sens e 23 2 of a bouque t o f circle s 24 0

projection 23 1 regular 26 6 space 23 1 trivial 23 1 universal 23 4

coverings equivalent 26 2 multiplicity o f 23 3 number o f sheet s o f 233 , 26 1 of Klei n bottl e 23 3 of basi c surface s 23 3 of a projectiv e spac e 23 2 of a toru s 23 2 of Mobiu s stri p 23 3

cube 11 0 de Morga n formula s 1 3 degree o f a poin t wit h respec t t o a loo p 25 1 diagonal 136 , 13 8 diameter o f a se t 2 1 difference o f set s 1 0 direct produc t 135 , see produc t discrete spac e 1 1 disjoint set s 8 disjoint su m 15 0 distance 18 , 4' 1

Gromov-Hausdorff 6 5 between set s 2 4

asymmetric 2 5 between a poin t an d a se t 2 3

edge 27 9 embedding

topological 74 , 12 1 isometric 6 5

element 3 maximal 3 8 minimal 3 8 greatest 3 8 smallest 3 8

elementary se t 13 6 epimorphism 18 4 equivalence clas s 14 1 equivalence relatio n 14 1 equivalent

embeddings 7 5 metrics 2 2 coverings 26 2

Euler characteristi c 28 8 exterior o f a se t 2 9 exterior poin t 2 9 factor

group 18 5 set 14 1 map 14 3

fiber of a produc t 13 5

finite intersectio n propert y 10 9 fixed poin t 98 , 25 4

Index 397

set 9 8 property 25 4

forest 29 2 function 5 5 fundamental grou p 21 5

of a bouque t o f circle s 23 8 of a cellula r spac e 295 , 29 8 of a finite spac e 216 , 24 1 of a len s spac e 29 9 of circl e 23 7 of a produc t 21 7 of a projectiv e spac e 23 8 of R n 21 6 of S n wit h n > 1 21 6 of a spher e wit h crosscap s an d handle s

297 of a topologica l grou p 21 8 of a toru s 23 7 of a kno t complemen t 30 0

fundamental theore m o f algebr a 24 8 G-set

left (right ) 19 6 homogeneous 19 6 19 8

generators o f a grou p 18 5 gluing 145 , 148 , 15 1 graph o f a ma p 13 6 group 90 , 18 1

Abelian 18 2 axioms 18 1 of homeomorphism s 19 5 fundamental 21 5 homology 1-dimensiona l 29 8 of a coverin g 26 1 operation 18 1 free 23 8 topological 18 7 trivial 18 2 cyclic 18 5

handle 72 , 15 2 Hasse diagra m 4 0 Hausdorff axio m 9 7 heriditary propert y 99 , 109 , 10 2 hierarchy o f covering s 26 2 Holder

inequality 1 9 map 6 2

homeomorphic 6 7 homeomorphism 6 7 homeomorphism proble m 7 3 homogeneous G-se t 19 6 homogeneous coordinate s 15 5 homomorphism

group 18 3 topological grou p 19 3 induced b y a continuou s ma p 24 7 induced b y a coverin g projectio n 26 1

A-homotopy 21 1

homotopy 20 8 free 211 , 216 stationary 21 1 rectilinear 20 9 of path s 21 1

homotopy class o f a ma p 20 9 equivalence 25 6

of cellula r space s 28 9 equivalent space s 25 6 groups 215 , 21 9

of a coverin g spac e 26 8 type o f a spac e 256 , 25 9

identity ma p 5 7 image

continuous 8 6 of a poin t 5 5 of a se t 5 6

inclusion (map ) 5 7 index

of a subgrou p 185 , 26 1 of a poin t wit h respec t t o a loo p 25 1

indiscrete spac e 1 1 induction

on compactnes s 11 3 on connectednes s 9 0

inequality triangle 1 8

ultrametric 2 5 Holder 1 9

injection 5 5 injective facto r

of a continuou s ma p 14 3 of a homomorphis m o f topologica l group s

193 interior

of a se t 2 9 operation o f 31 , 32 point 2 9

intermediate valu e theore m 89 , 24 9 intersection o f set s 8 Inverse Functio n Theore m 7 4 inverse ma p 5 7 irrational flow 19 7 isolated poin t 3 4 isometry 6 2 isomorphism 18 4

of topologica l group s 19 3 local 19 3

kernel o f a homomorphis m 18 4 Klein bottl e 73 , 14 9 knot 7 5 Kolmogorov axio m 10 0 Kuratowski proble m 3 1 Lagrange Theore m 18 5 Lebesgue Lemm a 11 2 lifting proble m 235 , 26 1

398 Index

limit of a sequenc e 9 8 point o f a se t 3 3

Lindelof Theore m 10 4 line

real 12 , 8 7 digital 4 1 with Ti-topolog y 1 2

list o f a coverin g 23 4 local

homeomorphism 23 3 isomorphism 19 3

loop 21 5 map 5 5

antimonotone 6 4 bijective 5 5 characteristic o f a cel l 28 1 closed 112 , 13 7 continuous 5 9

at a poin t 6 1 contractive 6 2 equi variant 19 6 factorization 14 1 graph o f 13 6 homotopy invers e 25 6 homotopy invertibl e 25 6 identity 5 7 image o f 5 6 inclusion 5 7 injective 5 5 inverse 5 7 invertible 5 7 locally bounde d 11 3 locally constan t 9 0 monotone 6 4 null-homotopic 20 9 one-to-one 5 5 open 13 7 proper 11 8 surjective 5 5

mapping 5 5 maximal To-quotien t o f a spac e 16 0 metric 1 8

pfr) 1 8 p-adic 2 5 equivalent 2 2 Euclidean 1 8 Hausdorff 2 4 of unifor m convergenc e 16 4 space 1 8

complete 11 5 Mobius ban d 14 8 Mobius stri p 14 8 monomorphism 18 4 multiplicity o f a coverin g 23 3 naive se t theor y 3 neighborhood

base a t a poin t 10 5 of a poin t 1 4 trivially covere d 23 1 symmetric 18 8

^-neighborhood o f a se t 11 1 e-net 11 4 norm 21 , 11 2 normalizer 19 6 notation

additive 18 2 multiplicative 18 2

one-point compactificatio n 11 8 open

set 11 , 1 3 in a subspac e 2 7

orbit 19 8 order

cyclic 42 , 4 2 linear 3 8 nonstrict partia l 3 5 strict partia l 3 5 of a grou p 18 5 of a grou p elemen t 18 5

p-adic number s 11 5 pantaloons 15 3 partition 83,14 1

closed 14 4 of unit y 12 0

to a cove r 12 0 of a se t 8 3 open 14 4

path 92 , 21 1 inverse 9 2 lifting homotop y theore m 23 5 lifting theore m 23 5 simple 29 3 stationary 9 2

path-connected cellular spac e 28 7 component 9 4 set 9 3 space 9 3

Peano curv e 6 6 plane

with hole s 7 2 with puncture s 7 2

Poincare grou p 21 5 point 1 1 polygon-connectedness 9 5 poset 3 6 preimage o f a se t 5 6 preorder 15 9 pretzel 15 3 product

free 30 1 with amalgamate d subgrou p 30 1

cellular space s 28 2

Index 399

fiber o f 13 5 of covering s 23 2 of homotop y classe s o f path s 21 2 of map s 138 , 13 8 of path s 92 , 21 2 of set s 13 5 of topologica l group s 19 4 of topologica l space s 136 , 13 9 semidirect 19 5

projection of a produc t t o a facto r 13 6 onto a quotien t spac e 14 1

projective plan e 15 0 projective spac e

real 15 5 complex 15 6 quaternionic 15 8

quaternion 15 7 quaternionic projectiv e lin e 15 8 quotient

group 18 5 set 14 1 space 14 2 map 14 3 topology 14 2

relation equivalence 14 1 linear orde r 3 8 nonstrict partia l orde r 3 5 reflexive 3 5 strict partia l orde r 3 5 transitive 35 , 15 9

restriction o f a ma p 5 8 retract 25 3

deformation 25 7 strong 25 7

retraction 25 3 deformation 25 7

strong 25 7 saturation o f a se t 14 1 skeleton o f a cellula r spac e 28 1 Seifert-van Kampe n Theore m 298 , 30 3 separation axio m

To (Kolmogoro v axiom ) 10 0 Ti (Tikhono v axiom ) 9 9 T2 (Hausdorf f axiom ) 9 7 T3 10 1 T4 10 1 in a topologica l grou p 18 9

sequentially compact 11 4 continuous 10 6

sequential closur e 10 6 set 3

algebraic 9 6 cardinality o f 10 3 connected 8 4

countable 10 3 compact 10 9 dense i n a se t 3 2 empty 5 everywhere dens e 3 2 bounded 2 1 closed 1 3 coincidence o f map s 9 8 convex 21 , 70 countable 10 3 cyclicly ordere d 4 2 fixed poin t 9 8 linearly ordere d 3 8 locally close d 3 4 locally finite 11 9 nowhere dens e 3 3 open 1 1 partially ordere d 3 6 path-connected 9 3 saturated 14 1 star-shaped 20 9

sets disjoint 8 of matrice s 96 , 11 1

simple pat h 29 3 simply connecte d spac e 21 7 simplicial

scheme 16 1 space 16 1

singleton 5 skeleton 28 1 space

arcwise connecte d 9 3 asymmetric 2 6 cellular 27 9

countable 28 1 finite 28 1 locally finite 28 1

compact 10 8 connected 8 3 contractible 25 8 covering 23 1 disconnected 8 3 discrete 1 1 finite 16 0 first countabl e 10 5 Hausdorff 9 7 indiscrete 1 1 locally compac t 11 7 locally contractibl e 26 4 locally path-connecte d 26 3 metric 1 8

complete 11 5 metrizable 22 , 10 7 micro simpl y connecte d 26 3 Niemyski 10 2 normal 10 1

400 Index

normed 2 1 of continuou s map s 16 3 of conve x figure s 11 6 of coset s 19 1 of simplice s 16 1 paracompact 11 9 path-connected 9 3 regular 10 1 second countabl e 10 4 separable 10 4 sequentially compac t 11 4 Sierpinski 1 2 simplicial 16 1 simply connecte d 21 7 smallest neighborhoo d 3 9 topological 1 1 totally disconnecte d 8 6 triangulated 16 1 ultrametric 2 5

spaces homeomorphic 6 7 homotopy equivalen t 25 6

sphere 1 9 with crosscup s 15 3 with handle s 15 3 with hole s 15 2

spheroid 21 9 subbase 1 7 subcover, subcoverin g 10 8 subgroup 18 4

isotropy 19 6 normal 18 5

of a topologica l grou p 19 2 of a topologica l grou p 19 1

submap 5 8 subordination o f covering s 26 2 subset 6

proper 6 subspace

cellular 28 1 of a metri c spac e 2 0 of a topologica l spac e 2 7

sum of set s 15 0 of space s 15 0

support o f a functio n 12 0 surjection 5 5 symmetric differenc e o f set s 1 0 Tietze Theore m 10 2 Tikhonov axio m 9 9 topological spac e 11 , see spac e topological structur e 1 1

coarser tha n anothe r on e 1 7 induced b y a metri c 2 2 interval 3 9 cellular 28 1 compact-open 16 3

finer tha n anothe r on e 1 7 left (right ) ra y 3 9 metric 2 2 of cycli c orde r 4 4 of pointwis e convergenc e 16 3 particular poin t 1 2 relative 2 7 subspace 2 7 poset 3 9

topological invarian t 7 3 topological propert y 7 3

hereditary 9 9 topology 11 , see topologica l structur e torus 14 0 translation (lef t o r right ) 18 8 translation alon g a pat h 22 0

for homotop y group s 22 2 in a topologica l grou p 22 2

tree 292 , 29 2 spanning 29 2

ultrametric 2 5 union o f set s 8 Urysohn Lemm a 10 2 Venn diagra m 9 vertex 27 9 winding numbe r 25 1

elementary topology - american mathematical society · the subject of the book: elementary topology...

Documents