engelking topology notes

7/24/2019 Engelking Topology Notes

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December 14, 2012

R. Engelking: General Topology

I started to make these notes from [E1] and only later the newer edition [E2] gotinto my hands. I dont think that there were too much changes in numberingbetween the two editions, but if youre citing some results from either of thesebooks, you should check the book, too.

Introduction

Algebra of sets. Functions

Cardinal numbers

For every cardinal number m, the number 2m, also denoted by exp m, is definedas the cardinality of the family of all subsets of a set X satisfying|X| = m.

To every well-ordered setXan ordinal number is assigned; it is called theorder type ofX.

Order relations. Ordinal numbers

Any ordinal number can be represented as + n where is a limit ordinalnumber and n N. The number + nis even (odd) ifnis even (odd).

A subset A of set X directed by is cofinal in X if for every xX thereexists an a

A such thatx

a. Cofinal subsets of linearly ordered sets and of

ordered sets are defined similarly.

The axiom of choice

Suppose we are given a set Xand a propertyPpertaining to subsets ofX; wesay thatP is a property offinite character if the empty set has this propertyand a set A Xhas propertyP if and only if all finite subsets ofA have thisproperty.

Lemma(Teichmuller-Tukey lemma). Suppose we are given a setXand a prop-ertyP of subsets of X. IfP is a property of finite character, then every setAX which has propertyP is contained in a setB X which has property

P and is maximal in the family of all subset ofXthat have

Pordered by

.

Real numbers

1 Topological spaces

1.1 Topological spaces. Open and closed sets. Bases. Clo-

sure and interior of a set

A family B O is called abase for a topological space (X, O) if every non-emptyopen subset ofXcan be represented as the union of a subfamily ofB.

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(B1) For anyU1, U2 Band every pointx U1 U2 there exists aU Bsuchthatx U U1 U2.

(B2) For everyx Xthere exists a U Bsuch that x U.If for some x Xand an open set U Xwe have x U, we say that U is

a neighbourhood ofx.A familyB(x) of neighbourhoods ofx is called a base for topological space

(X, O) at the pointx if for any neighbourhood V ofx there exists a U B(x)such that x U V.

The smallest cardinal number of the form|B|, whereBis a base for a topo-logical space (X, O), is called the weight of the topological space (X, O) and isdenoted by w(X, O).

A familyP O is called a subbase for a topological space (X, O) if thefamily of all finite intersections U1 . . . U k, where Ui P for i= 1, 2, . . . , k, is abase for (X, O).

base for topologybase at pointunion of bases at point = base for topologyThe character of a point x in a topological space (X, O) is defined as the

smallest cardinal number of the form|B(x)|, whereB(x) is a base for (X, O) atthe pointx; this cardinal number is denoted by(x, (X, O)). Thecharacter of atopological space(X, O) is defined as the supremum of all numbers (x, (X, O))forx X; this cardinal number is denoted by ((X, O)).

(X) 0=first-countablew(X) 0=second-countableLet (X, O) be a topological space and suppose that for every x Xa base

B(x) for (X,

O) atx is given; the collection

{B(x)

}xX is called aneighbourhood

system for the topological space(X, O). We shall show that any neighbourhoodsystem{B(x)}xX has the following properties:

(BP1) For everyx X,B(x) = and for every U B(x), x U.(BP2) Ifx U B(y), then there exists a V B(x) such thatV U.(BP3) For anyU1, U2 B(x) there exists a U B(x) such thatU U1 U2.

Corollary. (1.1.2) IfU is an open set andU A= , then also U A= .Theorem. (1.1.3) The closure operator has the following properties:

(CO1) = (CO2) A A(CO3) A B= A B

(CO4) (A) =A

Theorem. (1.1.5) For everyA Xwe haveInt A= X \ X\ A.Theorem. (1.1.6) The interior operator has the following properties:

(IO1) Int X=X

(IO2) Int A A

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(IO3) Int A B= Int A Int B(IO4) Int (Int A) = Int A

IfO1 andO2 are two topologies on X andO2 O1, then we say thattopologyO1 isfinerthan the topologyO2, or that topologyO2 iscoarser thanO1.

A family{As}sS of subsets of a topological space X is a locally finite iffor every point xXthere exists a neighbourhood such that the set{sS :U As= }is finite. If every point x Xhas a neighbourhood that intersectsat most one set of a given family, then we say that the family is discrete.

Theorem. (1.1.11) For every locally finite family {As}sSwe have the equality

sSAs =

sSAs.

Corollary. (1.1.12) LetFbe a locally finite family andF = F. If all mem-bers ofFare closed, thenF is a closed set and if all members ofFare clopen,thenF is clopen.

Theorem. (1.1.13) If{As}sSis locally finite (discrete), then the family{As}sSalso is locally finite (discrete).

Theorem. (1.1.14) Ifw(X) m, then for every family{Us}sSof open subsetsofXthere exists a setS0 Ssuch that|S0| m and

sS0

Us=sS

Us.

Theorem. (1.1.15) If w(X) m, then for every baseB for X there exists abaseB0 such that|B0| mandB0 B.Remark. (1.1.16) Let us note that in the proof of Theorem 1.1.14 we did notuse the fact that the members ofBare open (cf. the notion of network definedin Section 3.1).

Theory of real numbers (as equivalence classes) was proposed independentlyby Ch. Meray and G. Cantor.

(Exercise 1.1.C) A subset Uof a topological space satisfying the conditionU= Int Uis called an open domain.

1.2 Methods of generating topologies

Proposition. (1.2.1) Suppose we are given a setX and a familyBof subsetsof X which has properties (B1)-(B2). LetO be the family of all subsets of Xthat are unions of subfamilies of

B, i.e., let

U O if and only ifU=

B0 for a subfamilyB0 ofB.The familyO is a topology on X. The familyB is a base for the topologicalspace(X, O).Example. (1.2.2) Real numbers with topology defined by basea, b) = K -Sorgenfrey line.

Example. (1.2.4) L ={(x, y) R2 y 0}. For points of the liney = 0 wedefine bases by circles touching it and for y= 0 as usual. We get Niemytzkiplane.

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1.3 Boundary of a set and derived set. Dense and nowhere

dense sets. Borel setsBoundary ofA: Fr A= A X\ A= A \ Int ATheorem. (1.3.2) The boundary operator has the following properties:

(i) Int A= A \ Fr A(ii) A= A Fr A

(iii) Fr(A B) Fr A Fr B(iv) Fr(A B) Fr A Fr B(v) Fr(X\ A) = Fr A

(vi) X= Int A Fr A Int(X\ A)(vii) Fr A Fr A

(viii) Fr Int A Fr A(ix) A is open if and only ifFr A= A \ A(x) A is closed if and only ifFr A= A \ Int A

(xi) A is clopen if and only ifFr A= A point x in a topological space X is called an accumulation pointof a set

A

X if x

A

\ {x

}. The set of all accumulation points of A is called the

derived set ofA and is denoted by Ad.

Theorem. (1.3.4) The derived set has the following properties:

(i) A= A Ad

(ii) IfA B, thenAd Bd.(iii) (A B)d =Ad Bd

(iv)sS

Ads (sS

As)d

A set A X is called dense in X ifA= X.A set A

X is called co-dense in X ifX

\Ais dense.

A set A X is called nowhere dense in X ifA is co-dense.A set A X is called dense in itself ifA Ad.

Proposition. (1.3.5) The setA is dense inX if and only if every non-emptyopen subset ofXcontains points ofA.

The setA is co-dense inXif and only if every non-empty open subset ofXcontains points of complement ofA.

The setA is nowhere dense inXif and only if every non-empty open subsetofX contains a non-empty open set disjoint fromA.

Theorem. (1.3.6) If A is dense in X, then for every open U X we haveU=U A.

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The density of a spaceX is defined as the smallest cardinal number of the

form|A|, where A is a dense subset ofX. Ifd(X) 0, then we say that thespaceX is separable.Theorem. (1.3.7) For every topological spaceXwe haved(X) w(X).Corollary. (1.3.8) Every second-countable space is separable.

Borel sets, F, GComplement ofF set is G set.

1.4 Continuous mappings. Closed and open mappings.

Homeomorphisms

Proposition. (1.4.1) LetXandY be topological spaces andfa mapping ofX

to Y. The following conditions are equivalent:

(i) The mappingf is continuous.

(ii) Inverse images of all members of a subbase forY are open inX.

(iii) Inverse images of all members of a base forY are open inX.

(iv) There are neighborhood systems{B(x)}xX and{D(y)}yY forX andYrespectively, such that for every x X and V D(f(x)) there exists aU B(x) satisfyingf(U) V.

(v) For everyA Xwe havef(A) f(A).

(vi) For everyB Y we havef1(B) f1

(B).

(vii) For everyB Y we havef1(Int B) Int f1(B).Let us observe in connection with the above theorem, that iff: X Y then

for anyF (G)B Y the inverse imagef1(B) is anF-set (G-set). Inverseimage of Borel sets in Yare Borel sets in X. (1.4.G)

Theorem. (1.4.7) If a sequence (fi) of continuous functions from X to R orI is uniformly convergent to a real-valued function f, then f is a continuousfunction fromX to R. If allfis are functions to I, thenf: X I.Proposition. (1.4.8) Suppose we are given a setX, a family{Ys}sSof topo-logical spaces and a family of mappings

{fs

}sS, where fs is a mapping of X

to Ys. In the class of all topologies onX that makes all fss continuous thereexists a coarsest topology; this is the topologyOgenerated by the base consistingof all sets of the form

ki=1

f1si [Vi],

wheres1, s2, . . . , sk S andVi is an open subset ofYsi fori= 1, 2, . . . , k.The topologyO is called thetopology generated by the family of mappings

{fs}sS.

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Proposition. (1.4.9) A mapping f of a topological space X to a topological

spaceY whose topology is generated by a family of mappings{fs}sS, wherefsis a mapping of Y to Ys, is continuous if and only if the composition fsf iscontinuous for everys S.Theorem. (1.4.10) IfY is a continuous image ofX, thend(Y) d(X).Corollary. (1.4.11) Continuous images of separable spaces are separable.

A continuous mappingf: X Yis called aclosed (anopen) mapping if forevery closed (open) setA Xthe image f[A] is closed (open) in Y . Mappingswhich are simultaneously closed and open are called closed-and-openmappings.

Theorem. (1.4.12) A mapping f: X Y is closed (open) if and only if foreveryB Y and every open (closed) setA Xwhich containsf1(B), thereexists an open (a closed) setC

Y containingB and such thatf1(C)

A.

Theorem. (1.4.13) A mapping f: X Y is closed if and only if for everypoint yY and every open set U X which contains f1(y), there exists inY a neighbourhoodVof the pointy such thatf1(V) U.Theorem. (1.4.14) A mappingf: X Y is open if and only if there exists abaseB forXsuch thatf[U] is open inY for everyU B.Theorem. (1.4.16) For every open mapping f: X Y and every x X wehave (f(x), Y) (x, X). If, moreover, f[X] = Y, then w(Y) w(X) and(Y) (X).Example. (1.4.17) X =R, Y =R/N, f: X Y is closed and onto. We get(Y)>

0 and w(Y)>

0, whilew(X) =(X) =

0.

A() = space on a set with cardinality , topology= all subsets that do notcontainx0 and all subsets ofX that have finite complement.

1.5 Axioms of separations

Theorem. (1.5.1) For everyT0-spaceXwe have|X| exp w(X).Proposition. (1.5.2) Suppose we are given a setXand a collection{B(x)}xXof families of subsets of X which has properties (BP1)-(BP3). If in additionthe collection{B(x)}xX has the following property

(BP4) For every pair of distinct points x, y X there exist open set U B(x)andV

B(y) such thatU

V =

,

then the spaceXwith the topology generated by the neighbourhood system{B(x)}xXis a Hausdorff space.

Theorem. (1.5.3) For every Hausdorff space X we have|X| exp exp d(X)and|X| [d(X)](X).Theorem. (1.5.4) For any pairf, g of continuous mappings of a spaceX intoHausdorff spaceY the set{x X :f(x) =g(x)} is closed.

A topological spaceXis called aT3-spaceor regularspace, ifX is a T1-spaceand for every x Xand every closed set FX such that x / F there existopen sets U, V such that x U, F V andU V = .

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Proposition. (1.5.5) A T1-spaceX is a regular space if and only if for every

x X and every neighbourhood V of x in a fixed subbaseP there exists aneighbourhoodU ofx such thatU V.Theorem. (1.5.6) For every regular space we havew(X) exp d(X).

A topological space is called a T3 12 -space or Tychonoff space or completelyregular space, ifX isT1-space and for every x Xand every closed set F X,x / F there exists a continuous function f: X I such that f(x) = 0 andf(F) = 1.

A topological space is called T4-space or normal space, if X is a T1-spaceand for every pair of disjoint closed subsets A, B Xthere exist open sets U,V such that A U, B V andU V = .Theorem (Urysohns lemma). (1.5.10) For every pairA, B of disjoint closed

subsets of a normal spaceX there exists a continuous functionf: X I suchthatf(A) = 0 andf(B) = 1.

Corollary. (1.5.11) A subsetA of a normal spaceX is a closedG-set if andonly if there exists a continuous functionf: X Isuch thatA= f1(0).Corollary. (1.5.12) A subsetA of a normal spaceX is an openF-set if andonly if there exists a continuous functionf: X Isuch thatA= f1((0, 1).

Two subsetsAandBof a topological spaceXare calledcompletely separatedif there exists a continuous functionf: X Isuch thatf(A) = 0 andf(B) = 1.We say that f separates sets A and B .

A subset A of a topological space X is called functionally closed1 if A =

f1

(0) for some f: X I. Every functionally closed set is closed. The com-plement of functionally closed set is called functionally open.One readily verifies that a T1-space X is completely regular if and only if

the family of all functionally open sets is a base for X. In a normal spacefunctionally closed (open) sets coincide with closedG-sets (open F-sets).

Theorem. (1.5.13) Any disjoint functionally closed setsA, B in a topologicalspaceXare completely separated; moreover, there exists a continuous functionf:X Isuch thatA= f1(0) andB= f1(1).Lemma. (1.5.14) IfX is aT1-space and for every closed setF Xand everyopenW Xthat containsFthere exists a sequenceW1, W2, . . . of open subsetsof X such that F

i=1

Wi andWi

W for i = 1, 2, . . ., then the spaceX is

normal.

One can easily check that the condition in the above lemma is not onlysufficient but also necessary for normality of a T1-spaceX.

Theorem. (1.5.15) Every second-countable regular space is normal.

Theorem. (1.5.16) Every countable regular space is normal.

Example. (1.5.17) Sorgenfrey line Kis a normal space.

1The terms functionally closed set and functionally open set adopted here seem more

suitable than the terms zero-set and cozero-set which are generally used.

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A family {As}sSof subsets of a set Xis called acover ofX ifsSAs= X.IfXis a topological space and all setsAsare open (closed), we say that the cover{As}is open (closed). A family{As}sS is called point-finite (point-countable)if for every xX the set{sS :xAs} is finite (countable). Clearly everylocally finite cover is point-finite ().

Theorem. (1.5.18) For every point-finite open cover{Us}sSof a normal spaceX there exists an open cover{Vs}sSsuch thatVs Us for everys S.

A topological spaceX is aperfectly normal space ifXis a normal space andevery closed subset ofX is a G-set (equivalently every open subset is F).

Theorem(The Vedenissoff theorem). (1.5.19) For everyT1-space the followingconditions are equivalent:

(i) The spaceX is perfectly normal.

(ii) Open subsets ofX are functionally open.

(iii) Closed subsets ofX are functionally closed.

(iv) For every pair of disjoint closed subsetsA, B Xthere exists a continuousfunctionf: X Isuch thatf1(0) =A andf1(1) =B.

Theorem. (1.5.20) The class of allTi-spaces for i= 1 and4 and the class ofperfectly normal spaces are invariant under closed mappings.

1.5.C: A continuous mapping f: X X is called a retraction of X, iff f=f; the set of all values of a retraction ofX is called a retract ofX.

Any retract of a Hausdorff space is closed.

1.6 Convergence in topological spaces: Nets and filters.

Sequential spaces and Frechet spaces

We say that the net S ={x , } is finer than the net S={x, }it there exists a function of to with following properties:

(i) For every0 there exists a 0 such that () 0 whenever 0.

(ii) x() = x for .

A point x is called a cluster point of a net S ={

x

,

}

if for every0 there exists a 0 such that x U.Proposition. (1.6.1) Ifx is a cluster point of the netS that is finer thenS,thenx is a cluster point ofS. Ifx is a limit of S, then it is a limits ofS. Ifx is a cluster point of the netS, then it is a limit of some netS that is finerthanS.

Proposition. (1.6.3) The pointx belongs to A if and only if there exists a netconsisting of elements ofA and converging to X.

Corollary. (1.6.4) A set A is closed if and only if together with any net itcontains all its limits.

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Corollary. (1.6.5) The pointx belongs to Ad if and only if there exists a net

S ={x, } converging to X, such that x A and x= x for every .Proposition. (1.6.6) A mapping f of a topological space X to a topologicalspaceY is continuous if and only if

f[lim

x] lim

f(x)

for every net{x, } in the spaceX.Proposition. (1.6.7) A topological space X a Hausdorff space if and only ifevery net inXhas at most one limit.

Let R be a family of sets that contains together with A and Bthe intersectionA

B. By a filter in

R we mean a non-empty subfamily

F R satisfying the

following conditions:

(F1) / F(F2) IfA1, A2 F, then A1 A2 F.(F3) IfA F and A A1 R, then A1 R.

A filter-base inRis a non-empty familyG Rsuch that / G and(FB) IfA1, A2 G, then there exists an A3 G such that A3 A1 A2.

A pointxis called alimit of a filterFif every neighborhood ofxis a memberofF.

A pointx is called acluster point of a filterF ifx belongs to closure of everymember of

F.

We say that a filterF isfinerthan a filterF ifF F.Proposition. (1.6.8) Ifx is cluster point of the filterF that is finer thanF,thenx is a cluster point of the filterF. Ifx is a limit ofF, then it is a limitsofF. Ifx is a cluster point of the filterF, then it is a limit of some filterFthat is finer thanS.

Proposition. (1.6.9) The point x belongs to A if and only if there exists afilter-base consisting of subsets ofA converging to x.

Proposition. (1.6.10) A mapping f of a topological space X to a topologicalspaceY is continuous if and only if for every filter-baseG in the spaceX andthe filter-basef[G] = {f[A] :A G} in the spaceY we have

f[lim G] lim f[G].Proposition. (1.6.11) A topological spaceX is a Hausdorff space if and onlyif every filter inXhas at most one limit.

Theorem. (1.6.12) For every net S ={x, } in a topological space X,the familyF(S), consisting of all setsA Xwith the property that there existsa0 such thatx A whenever 0, is a filter in the spaceX and

lim F(S) = lim S.If a netS is finer than the netS, then the filterF(S)is finer than the filter

F(S).

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Theorem. (1.6.13) LetFbe a filter in a topological spaceX; let us denote by the set of all pairs(x, A), wherex A Fand let us define that(x1, A1) (x2, A2)ifA2 A1. The set is directed by, and for the netS(F) = {x, }, wherex =x for= (x, A) , we haveF= F(S(F)) and

lim S(F) = lim F.sequential space, Frechet space

Theorem. (1.6.14) Every first-countable space is a Frechet space and everyFrechet space is a sequential space.

Proposition. (1.6.15) A mapping f of a sequential space X to a topologicalspaceYis continuous if and only iff[lim xi] lim f(xi)for every sequence(xi)in the spaceX.

Proposition. (1.6.16) If every sequence in a topological spaceXhas at mostone limit, thenX is aT1-space. If, moreover, X is first-countable thenX is aHausdorff space.

Proposition. (1.6.17) A first-countable space X is a Hausdorff space if andonly if every sequence in the spaceXhas at most one limit.

1.7 Problems

1.7.1 Urysohn spaces and semiregular spaces I

TODO Urysohn space 2

A topological space X is called a semiregular space ifX is a T2-space andthe family of all open domains is a base for X.

Let (X, O) be a Hausdorff space. Generate on Xa topologyO

O by thebase consisting of all open domains of (X, O) and show that the space (X, O)is semiregular and has the same open domains as the space (X, O).

1.7.2 Cantor-Bendixson theorem

perfect set= dense in itself and closedscattered set= contains no non-empty dense in itself subset

Show that if each member of a familyAof subset of a space X is dense initself, then the union

A is dense in itself. Note that if A X is dense initself, then the closure A is dense in itself. Deduce from the above that everytopological space can be represented as the union of two disjoint sets, one ofwhich is perfect and the second one is scattered.

A point x of a topological space X is called a condensation point of a setA Xif every neighborhood ofx contains uncountably many points ofA; theset of all condensation points ofA is denoted by A0.

Verify that A0 Ad, A0 = A0 and (A B)0 = A0 B0. Show that forevery subset A of a second-countable space, the difference A \ A0 is countableand (A0)0 =A0.

Deduce from the above that every second-countable space can be representedas the union of two disjoint sets, of which one is perfect and the other countable(this is the Cantor-Bendixson theorem). Cantor and I. Bendixson proved thisfact independently in 1883 for subsets of the real line.

2TODO

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1.7.3 Cardinal functions I

The smallest infinite cardinal number such that every family of pairwise dis-joint non-empty open subsets of X has cardinality is called the Souslinnumber or cellularityand denoted byc(X).

w(X) d(X) c(X)The smallest infinite cardinal number such that every subset ofXconsist-

ing exclusively of isolated points (i.e. satisfying the equalityA = A \ Ad) hascardinality is denoted by hc(X).

w(X) hc(X) c(X)The smallest infinite cardinal number such that every closed subset con-

sisting exclusively of isolated points has cardinality is called the extent ofthe space Xand denoted by e(X).

w(X) hc(X) e(X)

For sake of simplicity, in all problems about cardinal functions, the cardinalfunctions defined in the main body of the book (weight, character and density,as yet) will be re-defined to assume only infinite values: the new value off(X)is defined to be0 if the old value is finite, and to be equal to the old value ifthis is an infinite cardinal number. (Sometimes topologists say that there areno finite cardinal numbers in general topology).

IfY is a continuous image ofX, then c(Y) c(X) and hc(Y) hc(X). If,moreover, X is a T1-space, then also e(Y)

e(X).

The tightness of a point x in a topological space Xis the smallest cardinalnumber m 0 with the property that ifx C, then there exists a C0 Csuch that|C0| m and x C0; this cardinal number is denoted by (x, X).The tightness of a topological spaceXis the supremum of all numbers (x, X)forx X; this cardinal number is denoted by (X).

(x, X) (x, X) and (X) (X).Tightness(X) is equal to the smallest cardinal number m 0 with the

property that for any C X which is not closed there exists a C0 C suchthat|C0| mand C0 \ C= .

For every sequential space we have (X) = 0.

2 Operations on topological spaces2.1 Subspaces

A= A M (A=in subspaceM)Proposition. (2.1.3) If the compositiongfof mappingsf: X Y andg : YZ is closed (open), then the restrictiong|f[X] :f[X] Z is closed (open).Proposition. (2.1.4) Iff: X Y is a closed (an open) mapping, then on anysubspace theL Y the restrictionfL : f1(L) L is closed (open).

homeomorphic embedding

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Theorem. (2.1.6) Any subspace of aTi-space is aTi-space fori 312 . Normal-ity is hereditary with respect to closed subsets. Perfect normality is a hereditaryproperty.

Two subsetsAandB of a topological spaceXare calledseparated ifAB= =AB. Two disjoint sets are separated if and only if neither of them containsaccumulation points of the other.

Theorem. (2.1.7) For everyT1-spaceXthe following conditions are equivalent:

(i) The spaceX is hereditarily normal.

(ii) Every open subspace ofX is normal.

(iii) For every pair of separated setsA, B X there exist open setsU, V Xsuch thatA U, B V andU V = .

Hereditarily normal spaces are sometimes called T5-spaces,and members ofthe narrower class of perfectly normal spaces are called T6-spaces.

Theorem (Tietze-Urysohn theorem). (2.1.8) Every continuous function froma closed subspaceMof a normal spaceX to I orR is continuously extendableoverX.

Theorem. (2.1.9) If a continuous mapping f of a dense subset A of a topo-logical spaceX to a Hausdorff spaceY is continuously extendable overX, thenthe extension is uniquely determined byf.

Niemytzki plane is not normal.

Proposition. (2.1.11) If

{Us

}sSis an open cover of a spaceX and

{fs

}sS,

wherefs : Us Y is a family of compatible continuous mappings, the combina-tionf= fs is a continuous mapping ofX to Y.Corollary. (2.1.12) A mappingfof a topological spaceXto a topological spaceY is continuous if and only if every point x X has a neighborhood Ux suchthatf|Ux is continuous.Proposition. (2.1.13) The same as preceding proposition for locally finite closedcover.

Theorem. (2.1.14) For every countable discrete family{Fi}i=1 of closed sub-sets of a normal spaceXthere exists a family{Ui}i=1of open subsets ofXsuchthatFi Ui fori= 1, 2, . . . andUi Uj = fori =j .Proposition. (2.1.15) Suppose we are given a topological space X, a cover{As}sS of the space X and a family{fs}sS of compatible mappings, wherefs : As Y such that the combinationf =

sSfs : X Y is continuous. If all

mappings fs are open (closed and the family fs[As] is locally finite), then thecombinationf is open (closed).

2.1.D: Verify that a subspace Mof a topological space X is a retract ofXif and only if every continuous mapping defined on M is extendable over Xof - equivalently if and only if there exists a mapping r : X M such thatr|M =idM.

2.1.E: Prove that normality is hereditary with respect to F-sets.2.1.I: Prove that the Sorgenfrey line is hereditarily separable.

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2.2 Sums

Theorem. (2.2.7) Any sum ofTi-spaces is aTi-space fori 6.

2.3 Cartesian products

Proposition. (2.3.1) The family of all setssS

Ws, whereWs is an open subset

of Xs and Ws= Xs only for finitely manys S, is a base for the Cartesianproduct

sS

Xs.

Moreover, if for every s S a baseBs for Xs is fixed, then the subfamilyconsisting of those

sS

Ws in whichWs Bs wheneverWs=Xs, also is a base.

The base forsS

Xs described in the first part of the above proposition is

called thecanonical basefor the Cartesian product.

Proposition. (2.3.2) If{Xs}sS is a family of topological spaces and As isfor every s S a subspace of Xs, then the two topologies defined on the setA =

sS

As, viz, the topology of the Cartesian product of subspaces{As}sSand the topology of a subspace of the Cartesian product

sS

Xs, coincide.

Proposition. (2.3.3) For every family of sets{As} where As Xs in theCartesian product

Xs we have

As=

As.

Corollary. (2.3.4) The set

As, where =As Xs, is closed in

Xs if and

only if everyAs is closed inXs.

Corollary. (2.3.5) The set

As , where = As Xs, is dense in

Xs ifand only if everyAs is dense inXs.

Projections are open but they arent closed in general.

Theorem. (2.3.11) Any Cartesian product ofTi-spaces is aTi-space fori 312 .If the Cartesian product

sS

Xs is a non-emptyTi-space, then allXss areTi-

spaces fori 6.Example. (2.3.12)K Kis not normal, K- the Sorgenfrey line.Theorem. (2.3.13) Ifw(Xs) 0 for every sS and card S thenw(

sSXs) .Similarly, if (Xs) 0 for every s S and card S then(

sSXs) .Corollary. (2.3.14) First-countability and second-countability are0-multiplicativeproperties.

Theorem (Hewitt-Marczewski-Pondiczery). (2.3.15) If d(Xs) 0 foreverys S and card S 2, thend(Xs) .Corollary. (2.3.16) Separability is ac-multiplicative property.

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Theorem. (2.3.17) If d(Xs) 0 for every s S, then any family ofpairwise disjoint non-empty open subsets of the Cartesian product has cardinality .Corollary. (2.3.18) In the Cartesian product of separable spaces any family ofpairwise disjoint non-empty open sets is countable.

Suppose we are given a topological space X, a family {Ys}sSof topologicalspaces and a family of mappingsF ={fs}, where fs : X Ys. We say thatthe familyF separates pointsif for every pair of distinct points x, y X thereexists a mapping fs F such that fs(x) =fs(y). If for everyx Xand everyclosed set F X such that x / F there exists a mapping fs F such thatfs(x) /fs(F), then we say that the familyF separates points and closed sets.Let us note that ifX is a T0-space, then every familyF separating points andclosed sets separates points as well.

Lemma. (2.3.19) If the mappingf: X Y is one-to-one and the one-elementfamily{f} separates points and closed sets, thenf is a homeomorphic embed-ding.

Theorem (The diagonal theorem). (2.3.20) If the familyF= {fs}sS, wherefs : X Ys, separates points, then the diagonal f =

sSfs : X

sS

Ys is

a one-to-one mapping. If, moreover, the familyF separates points and closedsets, thenf is a homeomorphic embedding.

In particular, if there exists ansSsuch thatfs is a homeomorphic em-bedding, thenf is a homeomorphic embedding.

Corollary. (2.3.21) IfXs= Xfor everys

S, then the diagonali =

idXs: XXs is a homeomorphic embedding; hence the diagonal of the Cartesian

productXm is homeomorphic to X.

By the graph of mapping fof a space X to a space Y, we mean the subsetof Cartesian productX Y defined by G(f) = {(x, y) X Y :y = f(x)}.Corollary. (2.3.22) For every continuous mappingf: X Y the graphG(f)is the image of X under the homeomorphic embedding idXf: X X Y.The restriction p|G(f) of the projection p : X YX is a homeomorphism.IfY is a Hausdorff space, thenG(f) is a closed subset ofX Y.

We say that the space X is universal for all spaces having a topologicalpropertyP ifXhas the propertyPand every space that has the propertyP isembeddable in X.

Theorem. (2.3.23) The Tychonoff cubeIm is universal for all Tychonoff spacesof weightm 0.

The Cantor cube of weightm 0 is the space Dm. The Cantor cube D0is called Cantor set. Cantor cube is universal space for all zero-dimensionalspaces of weight m.

Theorem. (2.3.24) For everym 0 and everyx Dm, we have(x, Dm) =m.

Corollary. (2.3.25) For everym 0 and everyx Im we have(x, Im) = m.

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TheAlexandroff cubeof weight 0is the spaceF, whereF is Sierpinskispace.Theorem. (2.3.26) The Alexandroff cube F is universal for all T0-spaces ofweight 0.Proposition. (2.3.27) If the Cartesian productf =

fs, wherefs : XsYs

andXs= fors S, is closed, then all mappingsfs are closed.The converse is not true in general.

Proposition. (2.3.29) The Cartesian product f =

fs, where fs : Xs YsandXs= fors S, is open if and only if all mappingsfs are open and thereexists a finite setS0 Ssuch thatfs(Xs) =Ys fors S\ S0.Proposition. (2.3.30) If mappingsf1, f2, . . . , f k, wherefi : Xi

Yi, are closed,

Y1is aT1-space andY2, Y3, . . . , Y kareT3-spaces, then the diagonalf=f1 . . . fkis closed.

Converse is not true in general. Proposition 2.3.30 cannot be generalized toinfinite diagonals.

Proposition. (2.3.32) If the diagonal f =fs is open, where fs : Xs Ys,then all mappingsfs are open.

The converse is not true, even for finite systems.

Proposition. (2.3.34) A netx in the Cartesian product

Xs converges to xif and only if everyps(x) converges to ps(x).

Proposition. (2.3.35) If

Fis a filter in the Cartesian productXs, then for

every s S the familyFs ={ps(F) : F F} is a filter in Xs. The filterFconverges to x if and only if the filterFs converges to ps(x) for everys S.Example. (2.3.36) Normality is not a hereditary property.

Example. (2.3.B) Int(A B) = Int A Int B, Fr(A B) = Fr A Fr BIfAs is an F-set (G-set) and|S| 0, then

As is and F-set (G-set).

Example. (2.3.C)Xis Hausdorff if and only if the diagonal of the Cartesianproduct X Xis closed in X X.Example. (2.3.L) If a topological property P is hereditary with respect toboth closed subsets and open subsets and is countably multiplicative, then, inthe class of Hausdorff spaces, Pis hereditary with respect to G sets.

If a topological propertyPis hereditary with respect to both closed subsetsand open subsets and is multiplicative, then if the closed interval I has P, allTychonoff spaces have P.

2.4 Quotient spaces and quotient mappings

Proposition. (2.4.2) A mapping f of a quotient space X/E to a topologicalspaceYis continuous if and only if the compositionf q is continuous.

Letf: X Ybe continuous. Let E(f) be equivalence relation on Xdeter-mined by f. The mapping f: X Y can be represented as the compositionf q,f is continuous.

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Proposition. (2.4.3) For a mappingf of a topological spaceXonto a topolog-

ical spaceY the following conditions are equivalent:(i) The mappingf is quotient.

(ii) The setf1(U) is open inXif and only ifU is open inY.

(iii) The setf1(F) is closed inX if and only ifF is closed inY.

(iv) The mappingf: X/E(f) Y is homeomorphism.Corollary. (2.4.4) The composition of two quotients mapping is a quotientmapping.

Corollary. (2.4.5) If the composition gf of two mappings is quotient, then gis a quotient mapping.

Corollary. (2.4.6) If for a continuous mapping f: X Y there exists a setA X such that f(A) = Y and the restriction f|A is quotient, then f is aquotient mapping.

Corollary. (2.4.7) Every one-to-one quotient mapping is a homeomorphism.

Corollary. (2.4.8) Closed mappings onto and open mappings onto are quotientmappings.

Proposition. (2.4.9) For an equivalence relation E on a topological space Xthe following conditions are equivalent:

(i) The natural mappingq: X

X/E is closed (open).

(ii) For every closed (open) set A X the union of all equivalence classesthat meetA is closed (open) inX.

(iii) For every open (closed) set A X the union of all equivalence classesthat are contained inA is open (closed) inX.

Corollary. (2.4.10) The quotient mappingf: X Y is closed (open) if andonly if the setf1f(A) X is closed (open) for every closed (open) A X.

We say that an equivalence relation Eon a space X isclosed (open) equiva-lence relationif the natural mapping g : X X/Eis closed (open). Decompo-sitions of topological space that correspond to closed (open) equivalence relationare calledupper (lower semicontinuous). In this context the wordidentificationis also often used, mainly with respect to upper semicontinuous decompositions:we say that the quotient space X/E, whereE is the equivalence relation corre-sponding to the decompositionE, is obtained by identifying each element ofEto a point.

adjunction space = we are given two disjoint topological spaces X and Yand a continuous mappingf: M Ydefined on a closed subsetMof the spaceX. Adjunction space = (X Y)/E.Theorem. (2.4.13) If M is a closed subspace of X andE is an upper semi-continuous decomposition of M, then the decomposition ofX into elements ofE and one-points set{x} withx X\ M is upper semicontinuous.

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Proposition. (2.4.14) A quotient space of a quotient space ofX is a quotient

space ofX. More precisely . . .Proposition. (2.4.15) If f: X Y is a quotient mapping, then for any setB Y which is either closed or open, the restriction fB : f1(B) B is aquotient mapping.

In other words, ifE is an equivalence relation on a space X, then for anyA Xwhich is either open or closed and satisfies the conditionq1q(A) =A,whereq is the natural mapping, the mappingq|A : A/(E|A) q[A]X/E ishomeomorphism.

Proposition. (2.4.18) Suppose we are given a topological space X, a cover{As}sS of the space X and a family{fs}sS of compatible mappings, wherefs : As Ysuch that the combinationf= fs : X Y is continuous. If thereexists a set S0 Ssuch that the restriction fs|As : As fs(As) are quotientfor s S0 and{fs(As)}sS0 is either an open cover of Y or a locally finiteclosed cover ofY, then the combinationfis a quotient mapping.

Now, suppose we are given a family{Xs}sSof topological spaces and forevery sS an equivalence relation Es on Xs. Letting{xs}E{ys} if and onlyif xsEsys for every s S we define an equivalence relation E on the Carte-sian product

sS

Xs; this relation is called the Cartesian product of relations

{Es}sS.Proposition. (2.4.19) If for every s S, Es is an open equivalence relationon a spaceXs andqs : Xs Xs/Es is the natural mapping, then the mappingsS

qs :sS

Xs/ sS

Es

sS(Xs/Es) is a homeomorphism.

Example. (2.4.20) Two quotient maps such their product is not quotient. X1=Y1= R\{ 12 , 13 , . . .} andf1= idX1 . X2= R,Y2= R/N,f2 : X2 Y2is a naturalmapping. f=f1 f2 is not a quotient mapping.Example. (2.4.E) Sumfs is quotient if and only if all mappings fs are quo-tient.

For every retractionf: X Xthe restrictionf|X: X f(X) is a quotientmapping.

2.4.F: f: X Y if X onto Y is called hereditarily quotient if for everyB Y the restriction fB : f1(B) B is a quotient mapping.

A mappingf: X

Y ofXontoY is hereditarily quotient if and only if the

setf[f1(B)] Y is closed for every B Y or equivalently - if and only if forevery y Yand any open U X that containsf1(y), we have y Int f[U].

Composition of two hereditarily quotient mappings is a hereditarily quo-tient mapping. Sum of hereditarily quotient mappings is a hereditarily quotientmapping. Proposition 2.4.18 holds also for hereditarily quotient mappings.

Any quotient mapping f: X Y onto a Frechet space Y in which everysequence has at most one limit (in particular, onto a FrechetT2-space) is hered-itarily quotient.

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2.5 Limits of inverse systems

Suppose that to every in a set directed by the relation corresponds atopological spaceX, and that for any a continuous mapping : XX is defined; suppose further that

=

for and that =

idX . In this situation we say that the family S ={X, , } is an inversesystemof the spaces X; the mappings

are called bonding mappingsof the

inverse system S. If =Nwith natural order, S is called inverse sequence.Let S be an inverse system; an element{x} of the Cartesian product

X is called atthreadofS if (x) =xfor any , and the subspace

of

X consisting of all threads of S is called limit of the inverse system andis denoted by lim S.Proposition. (2.5.1) Limit of an inverse system of Hausdorff spaces is a closedsubset of the Cartesian product.

Proposition. (2.5.2) The limit of an inverse system ofTi-spaces is aTi-spacefori 31

2.

Example. (2.5.3) Suppose we are given a family {Xs}sSof topological spaceswhere|S| 0. Observe that the family of all finite subsets ofS is directedby inclusion. Letting X =

sXs we obtain inverse system. (

is the

restriction of elements ofX to the subset of the set .) Limit of this systemis cartesian product

sSXs.

Let X= lim S. A mapping =p|X: X X is called the projection ofthe limit ofS to X.

Proposition. (2.5.5) The family of all sets 1 (U), where U is an open

subset ofX and runs over a subset cofinal in, is a base for the limit ofthe inverse system S.

Moreover, if for every a baseB forX is fixed, then the subfamilyconsisting of those1 (U) in whichU B, also is a base.Proposition. (2.5.6) For every subspaceA of the limitXof an inverse systemS= {X, , } the family SA = {A, , }, whereA =[A] and (x) = (x) forx A, is an inverse system and lim SA= A X.Corollary. (2.5.7) Any closed subspaceA of the limitXof an inverse systemS ={X, , } is the limit of the inverse system SA ={A, , } of closedsubspacesA of the spacesX.

Theorem. (2.5.8) LetPbe a topological property that is hereditary with respectto closed subsets and finitely multiplicative. A topological spaceXis homeomor-phic to the limit of an inverse system ofT2-spaces with the propertyPif and onlyifXis homeomorphic to a closed subspace of a Cartesian product ofT2-spaceswith the propertyP.

Suppose we are given two inverse systems S= {X, , } and S = {Y ,

, };

a mapping of the system S to the system S is a family{, f} consisting of anondecreasing function from to such that the set [] is cofinal in ,and of continuous mappings f: X() Y such that

f =f()() ,

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i.e., such that the diagram

X()f

()

()

Y

X()

f

Y

is commutative for any , satisfying .Any mapping of an inverse system S to an inverse system S induces a

continuous mapping of lim Sto lim S. This mapping is called thelimit mappinginduced by{, f} and is denoted by lim{, f}.Lemma. (2.5.9) Let{, f} be a mapping of an inverse systemS to an inversesystemS

. If all mappingsf are one-to-one, the limit mappingf= lim{, f}also is one-to-one. If, moreover, al l mappingsf are onto, falso is a mappingonto.

Proposition. (2.5.10) Let{, f} be a mapping of an inverse systemS to aninverse systemS. If all mappingsf are homeomorphisms, the limit mappingf= lim{, f} also is a homeomorphism.Corollary. (2.5.11) LetS= {X, , } be an inverse system and a subsetcofinal in. The mapping consisting in restricting all threads fromX= lim Sto is a homeomorphism of X onto the space X = lim S

, where S =

{X,

, }.

Corollary. (2.5.12) LetS =

{X, ,

}be an inverse system; if in the directed

set there exists and element 0 such that 0 for every , then thelimit ofS is homeomorphic to the spaceX0 .

Theorem. (2.5.13) For every mapping{, f} of an inverse systemS = {X, , }to an inverse systemS = {Y , , }there exists a homeomorphic embeddingh : lim S

Z, whereZ =X(), such that lim{, f} = (

f)h.

If allX() are Hausdorff spaces, thenf[lim S] is a closed subset of

Z.

Theorem. (2.5.14) For every inverse systemS = {X, , } and any0 there exist an inverse system S ={Y , , }, whereY =X0 for all , a homeomorphism h : lim S

X0 , and a mapping{, f} of S to S,wheref are bonding mappings of S, such that0 =h lim

{, f

}.

2.6 Function spaces I

YX = the set of all continuous mappings from X to Ytopology of uniform convergence

Proposition. (2.6.2) For every topological spaceX the setIX is closed in thespaceRX with the topology of uniform convergence.

Now let X and Y be arbitrary topological spaces; for A X and B Ydefine

M(A, B) = {f YX ; f[A] B}. (1)

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Denote byF the family of all finite subsets of X and letO be the topologyof Y. The familyBof all sets

ki=1

M(Ai, Ui)), where Ai F and Ui O fori= 1, 2, . . . , k, generates a topology on YX ; this topology is called the topologyof pointwise convergenceonYX . The familyBis a base for the space YX withthe topology of pointwise convergence.

Proposition. (2.6.3) The topology of pointwise convergence onYX coincideswith the topology of a subspace of the cartesian product

xXYx, whereYx = Y

for everyx X.Theorem. (2.6.4) If Y is a Ti-space, then the space Y

X with the topology ofpointwise convergence also is aTi-space fori 312 .Proposition. (2.6.5) A net

{f;

} in the spaceYX with the topology of

pointwise convergence converges tof YX if and only if the net{f(x), }converges to f(x) for everyx X.Proposition. (2.6.6) For every topological space X the topology of uniformconvergence onRX is finer than the topology of pointwise convergence.

Proposition. (2.6.9) For every family{Xs}sSof non-empty topological spacesand a topological spaceY, the combination :

sS(YXs) Y

(

sS

Xs)is a home-

omorphism with respect to the topology of pointwise convergence on functionspaces.

Proposition. (2.6.10) For every topological spaceX and a family{Ys}sS oftopological spaces, the diagonal :sS(YXs ) (sSYs)X is a homeomorphismwith respect to the topology of pointwise convergence on function spaces.

Let us observe that any mappingsg : Y Zandh : T Xinduce mappingg ofYX to ZX and h ofYX to YT defined by letting

g(f) =gf forf YX and h(f) =f h for f YX . (10)Since

1g (M(A, B)) =M(A, g1(B)) and 1h (M(A, B)) =M(h[A], B), (11)

both g and h are continuous with respect to the topology of pointwise con-vergence on function spaces.

The mappings g and hare connected with the operation of compositionof mappings; in fact from (10) it follows immediately that

g(f) = (g, f) and h(f) = (f, h).

The mapping of YX X to Y defined by (f, x) = f(x) is called theevaluation mapping ofYX . It is also connected with the operation ; namely, is the composition of mappings

YX XidYXiXYX X{p} Y{p} i1Y Y,i.e. = i1Y (idYX iX)

(12)

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One easily sees that the formula

{[(f)](z)}(x) =f(z, x), (13)

where f is a mapping of Z X to Y, defines a one-to-one correspondence between the set of all (not necessarily continuous) mapping of ZXto Yand theset of all mapping ofZto the set of all mapping ofXtoY; this correspondenceis called the exponential mapping.

We say that a topology on YX is proper if for every space Zand any fY(ZX) the mapping (f) belongs to (YX)Z. Similarly, we say that a topologyonYX isadmissibleif for every spaceZand anyg (YX)Z the mapping 1(g)belongs to Y(ZX). A topology onYX that is both proper and admissible iscalled an acceptable topology.

Proposition. (2.6.11) A topology onY

X

is admissible if and only if the eval-uation mapping ofYX is continuous, i.e., if :YX X Y.Proposition. (2.6.12) For every pairX, Y of topological spaces and any twotopologiesO,O on the function spaceYX we have:

(i) If the topologyO is proper andO O, the topologyO is proper.(ii) If the topologyOis admissible andO O, then the topologyOis admis-

sible.

(iii) If the topologyO is proper and the topology O is admissible, thenO O.(iv) OnYX there exists at most one acceptable topology.

The topology of pointwise convergence is proper.The topology of pointwise convergence is generally not admissible; indeed for

this topology the fact that g is in (YX)Z means that for allz0 Z andx0 Xthe mapping [g(z0)](x) and [g(z)](x0) are continuous, while the fact that

1(g)is in Y(ZX) means that g is continuous with respect to both coordinates.

The topology of uniform convergence is admissible. On the other hand, thetopology of uniform convergence is generally not proper.

2.7 Problems

2.7.1 Cardinal functions II

fis cardinal function

hfis supremum over all subspaces. Hereditary density,

hereditary Souslin number etc.hw(X) =w(X), h(X) =(X), h(X) =(X), hc(X) =he(X)hd(X) (X)IfA is a dense subspace ofX, thenc(A) =c(X), but not necessarilyd(A)

d(X).R with the topology generated by the base (a, b) \ A, where|A| 0, is a

Hausdorff space such thathd(X)> hc(X). The existence of such regular spaceis connected with Souslins problem.

Souslins problem - the question whether there exists a linearly ordered spaceX such that c(X) =0 and d(X) >0 (a Souslin space). If X is a Souslinspace then c(X X)> 0.

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2.7.2 Spaces of closed subsets I

2.7.20(a) For any topological space Xwe denote by 2X the family of all non-empty closed subsets ofX. The family Bof all the sets of the form V(U1, . . . , U k) ={B 2X : B

ki=1

Ui andB Ui=for i = 1, 2, . . . , k}, where U1, . . . , U k isa sequence of open subsets ofX generates a topology on 2X ; this topology iscalled the Vietoris topology on 2X and the set 2X with the Vietoris topology iscalled the exponential space ofX.

3 Compact spaces

3.1 Compact spaces

Let us recall that a cover of a set X is a family{As}sSof subsets ofX suchthat

sSAs =X, and that - ifX is a topological space-{As}sS is an open

(a closed) cover of X if all sets As are open (closed). We say that a coverB={Bt}tT is a refinement of another coverA={As}sSof the same set Xif for everyt Tthere exists ans(t) Ssuch thatBt As(t); in this situationwe say also thatB refinesA. A coverA ={As}sS of X is a subcover ofanother coverA ={As}sS ofX ifS S and As = As for every sS. Inparticular, any subcover is a refinement.

A topological space X is called a compact space if X is a Hausdorff spaceand every open cover of Xhas a finite subcover, i.e., if for every open cover{Us}sS of the space X there exists a finite set{s1, s2, . . . , sk} Ssuch thatX=Us1 Us2 . . . Usk .Theorem. (3.1.1) A Hausdorff spaceX is compact if and only if every familyof closed subsets ofX which has the finite intersection property has non-emptyintersection.

Theorem. (3.1.2) Every closed subspace of a compact space is compact.

Theorem. (3.1.3) If a subspaceA of a topological spaceXis compact, then forevery family{Us}sSof open subsets ofXsuch thatA

sS

Us there exists a

finite set{s1, . . . , sk} Ssuch thatA ki=1

Usi .

Corollary. (3.1.4) Let X be a Hausdorff space and{F1, . . . , F k} a family ofclosed subsets ofX. The subspaceF =

ki=1

Fi ofX is compact if and only if all

subspacesFi are compact.

Corollary. (3.1.5) Let U be an open subset of a topological space X. If a family{Fs}sS of closed subsets of X contains at least one compact set - inparticular, if X is compact - and if

sS

Fs U, then there exists a finite set

{s1, . . . , sk} Ssuch thatki=1

Fsi U.

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Theorem. (3.1.6) If A is a compact subspace of a regular space X, then for

every closed subsetB X\ A there exist open setsU, V X such thatA U,B V andU V = .If, moreover, B is a compact subspace ofX, then it suffices to assume that

X is a Hausdorff space.

Theorem. (3.1.7) IfA is a compact subspace of a Tychonoff spaceX, then forevery closed set B X\ A there exists a continuous functionf: X I suchthatf(x) = 0 forx A andf(x) = 1 forx B.Theorem. (3.1.8) Every compact subspace of a Hausdorff spaceX is a closedsubset ofX.

Theorem. (3.1.9) Every compact space is normal.

Theorem. (3.1.10) If there exists a continuous mappingf: X

Y of a com-pact spaceX onto a Hausdorff spaceY, thenY is a compact space.

In other words, a continuous image of a compact space is compact, providedit is a Hausdorff space.

Corollary. (3.1.11) Iff: X Y is a continuous mapping of a compact spaceX to a Hausdorff spaceY, thenf[A] =f[A] for everyA X.Theorem. (3.1.12) Every continuous mapping of a compact space to a Haus-dorff space is closed.

Theorem. (3.1.13) Every continuous one-to-one mapping of a compact spaceonto a Hausdorff space is a homeomorphism.

Corollary. (3.1.14) Let

O1 and

O2 be two topologies defined on a setXand let

O1 be finer thanO2. If the space(X, O1)is compact and(X, O2)is a Hausdorffspace, thenO1= O2.

In other words, among all Hausdorff topologies, compact topologies are min-imal.

Lemma. (3.1.15) IfA is a compact subspace of a spaceX andy a point of aspace Y, then for every open set W X Y containing A {y} there existopen setsU X andV Y such thatA {y} U V W.Theorem (The Kuratowski theorem). (3.1.16) For a Hausdorff space X thefollowing conditions are equivalent:

(i) The spaceX is compact.

(ii) For every topological spaceY the projectionp : X Y Y is closed.(iii) For every normal spaceY the projectionp : X Y Y is closed.

A familyN ={Ms}sSof subsets of a topological space X is a networkfor X if for every point x X and any neighborhood U if x there exists ans Ssuch that x Ms U. Clearly, any base for X is a network for X: itis a network of a special kind, one whose members all are open. Thenetworkweightof a space Xis defined as the smallest cardinal number of the form|N|,whereN is a network for X. Clearly, for every topological space X we havenw(X) w(X), nw(X) |X| andd(X) nw(X). For every T0-space we have|X| exp nw(X).

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Lemma. (3.1.18) For every Hausdorff spaceXthere exists a continuous one-

to-one mapping ofX onto a Hausdorff spaceY such thatw(Y) nw(X).Theorem. (3.1.19) For every compact spaceXwe havenw(X) =w(X).

Corollary. (3.1.20) If a compact spaceXhas a cover{As}sSsuch thatw(As) m 0 fors S and|S| m, thenw(X) m.Theorem. (3.1.21) For every compact spaceXwe havew(X) |X|.Theorem. (3.1.22) If a compact spaceY is a continuous image of a spaceX,thenw(Y) w(X).Theorem. (3.1.23) A Hausdorff spaceXis compact if and only if every net inXhas a cluster point.

The filter counterpart of the above theorem reads as follows:

Theorem. (3.1.24) A Hausdorff spaceX is compact if and only if every filterinXhas a cluster point.

Example. (3.1.26) X = C1 C2 - two concentric circle, the projection ofC1onto C2 from the point (0, 0) will be denoted by p. On the set X we shallgenerate a topology by defining a neighbourhood system {B(z)}zX ; namelyletB(z) ={z} for z C2 and for z C1 letB(z) ={Uj(z)}j=1, whereUj =Vj p[Vj \ {z}] andVj is the arc ofC1 with centre at z and of length 1/j.

The space Xis called the Alexandroff double circle.Xis a compact space

Example. (3.1.27) W = 1+ 1, base (y, x

and{

0}

. W is a compact space.W0= W\{1} - subspace. Every continuous function f: W0 I is extendableover W (every such a function is eventually constant). W0 is not perfectly nor-mal. Wis hereditarily normal but not perfectly normal. W0 is first countable.Wis not a sequential space and it has no countable base at 1.

Example. (3.1.28) Cantor set D0 is homeomorphic to a subspace of the real

line. C= sets of all numbers of the formi=1

2xi3i , where xi {0, 1} for i= 1, 2, . . .

We putf(x) = {xi}, f is a homeomorphism.Theorem. (3.1.29) For every infinite compact spaceXwe have|X| exp (X).Corollary. (3.1.30) Every first countable compact space has cardinality c.

A topological spaceXis called aquasi-compact spaceif every open cover ofXhas a finite subcover. The reader can easily verify that Theorems 3.1.1-3.1.3,Corollaries 3.1.4-3.1.5, Theorem 3.1.10, 3.1.16, 3.1.23 and 3.1.24 of this section,as well as Theorems 3.2.3, 3.2.4, and 3.2.10 of the next section, remain valid,along with their proofs, when one replaces compact by quasi-compact andHausdorff space by topological space.

3.1.F: The pseudocharacter of a point x in a T1-space X is defined as thesmallest cardinal number of the form|U |, whereU is a family of open subsetsofX such that

U ={x}; this cardinal number is denoted by (x, X). Thepseudocharacter of a T1-space X is defined as the supremum of all numbers(x, X) for all x X; this cardinal number is denoted by (X).

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For every T1-space X we have (x, X) (x, X) and (X) (X). If X is a compact space then (x, X) = (x, X) and (X) = (X). For everyHausdorff space X we have (X) exp d(X). For every regular space X wehave|X| exp[d(X)(X)].

3.2 Operations on compact spaces

Theorem. (3.2.1) LetA be a dense subspace of a topological spaceX andf acontinuous mapping ofAto a compact spaceY. The mappingfhas a continuousextension overXif and only if for every pairB1,B2of disjoint closed subsets ofY the inverse imagesf1(B1) andf

1(B2) have disjoint closures in the spaceX.

Theorem. (3.2.2) Every compact space of weightm 0is a continuous imageof a closed subspace of the Cantor cubeD

m

.

Theorem. (3.2.3) The sumsS

Xs, whereXs= fors S, is compact if andonly if all spacesXs are compact and the setS is finite.

Theorem (The Tychonoff theorem). (3.2.4) The Cartesian productsS

Xs,

whereXs= fors S, is compact if and only if all spacesXs are compact.Theorem. (3.2.5) The Tychonoff cube Im is universal for all compact spacesof weightm 0.Theorem. (3.2.6) A spaceXis a Tychonoff space if and only if it is embeddablein a compact space.

Theorem. (3.2.8) A subspace A of Euclidean n-space Rn is compact if andonly if the setA is closed and bounded.

Corollary. (3.2.9) Every continuous real-valued function defined on a compactspace is bounded and attains its bounds.

Theorem (The Wallace theorem). (3.2.10) If As is a compact subspace of atopological spaceXs forsS, then for every open subset W of the Cartesianproduct

sS

Xs which contains the setsS

As there exist open setsUs Xs suchthatUs=Xs for only finitely manys S and

sS

AssS

Us W.

Theorem (The Alexandroff theorem). (3.2.11) For every closed equivalencerelationE on a compact spaceX there exists exactly one (up to a homeomor-phism) Hausdorff space Y and a continuous mapping f: X Y of X ontoY such that E = E(f), viz. the quotient space X/Eand the natural quotientmappingq: X X/E; moreoverY is a compact space.

Conversely, for every continuous mappingf : X Yof a compact spaceXonto a Hausdorff spaceY the equivalence relationE(f) is closed.

Theorem. (3.2.13) The limit of an inverse system S ={X, , } of non-empty compact spaces is compact and non-empty.

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Theorem. (3.2.14) Let{, f} be a mapping of an inverse systemS = {X, , }of compact spaces to an inverse system S

={Y ,

,

} of T1-spaces. If allmappings f are onto, the limit mapping f = lim{, f} also is a mappingonto.

Corollary. (3.2.15) If in an inverse systemS = {X, , }of compact spacesall bonding mappings are onto, then the projections : lim S X alsoare mappings onto.

Corollary. (3.2.16) If S ={X, , }, where=, is an inverse system ofT1-spaces,X is a compact space, and{f} wheref : X X, is a familyof mappings onto such that f = f for any, satisfying , thenthe limit mapping lim f also is a mapping onto.Corollary. (3.2.17) If S ={X, , }, where=, is an inverse system ofcompact spaces, X is aT1-space, and{f} wheref : X X, is a familyof mappings onto such thatf

= f for any, satisfying , then

the limit mapping lim f also is a mapping onto.Lemma (The Dini theorem). (3.2.18) Let X be a compact space and{fi} asequence of continuous real-valued functions defined onXand satisfyingfi(x) fi+1(x) for all x X and i = 1, 2, . . . If there exists a function f RX suchthatf(x) = lim fi(x) for everyx X, thenf= lim fi, i.e. the sequence{fi} isuniformly convergent to f.

Lemma. (3.2.19) There exists a sequence{wi} of polynomials which is uni-formly convergent to the function

t on the closed intervalI.

Lemma. (3.2.20) LetPbe a ring of continuous and bounded real-valued func-

tions defined on a topological spaceX. If the ringP contains all constant func-tions and is closed with respect to uniform convergence, then for everyf, g Pthe functionsmax(f, g) andmin(f, g) belong to P.

Theorem(The Stone-Weierstrass theorem). (3.2.21) If a ringPof continuousreal-valued functions defined on a compact spaceX contains all constant func-tions, separates points and is closed with respect to uniform convergence (i.e., isa closed subset of the spaceRX with the topology of uniform convergence), thenP coincides with the ring of all continuous real-valued functions onX.

3.3 Locally compact spaces and k-spaces

A topological spaceXis called a locally compact spaceif for every x X thereexists a neighbourhood Uof the point x such that U is a compact subspace ofX.

Theorem. (3.3.1) Every locally compact space is a Tychonoff space.

Theorem. (3.3.2) For every compact subspaceA of a locally compact spaceXand every open setV Xthat containsA there exists an open setU X suchthatA U U V andU is compact.Corollary. (3.3.3) For every compact subspace A of a locally compact spaceX and every open set V that contains A there exists a continuous functionf: X I such thatf(x) = 0 forx A, f(x) = 1 forxX\ V and the setf1(0, a) is compact for everya


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Theorem. (3.3.4) The character of a pointx in a locally compact spaceX is

equal to the smallest cardinal number of the form|U |, whereU is a family ofopen subsets ofXsuch thatU= {x}.Theorem. (3.3.5) For every locally compact spaceXwe havenw(X) =w(X).

Corollary. (3.3.6) For every locally compact spaceXwe havew(X) |X|.Corollary. (3.3.7) If a locally compact spaceYis a continuous image of a spaceX, thenw(Y) w(X).Theorem. (3.3.8) If X is a locally compact space, then every subspace of Xthat can be represented in the form F V, where F is closed in X and V isopen inX, also is locally compact.

Theorem. (3.3.9) A locally compact subspace M of a Hausdorff space X isan open subset of the closure M of the set M in the space X, i.e., it can be

represented in the formF V, whereF is closed inX andV is open inX.Corollary. (3.3.10) A subspaceMof a locally compact spaceX is locally com-pact if and only if it can be represented in the formF V, whereF is closed inX andV is open inX.

Corollary. (3.3.11) A topological space is locally compact if and only if it ishomeomorphic to an open subspace of a compact space.


Xs is locally compact if and only if all spaces

Xs are locally compact.

Theorem. (3.3.13) The Cartesian product

sSXs, whereXs= fors S,is locally compact if and only if all spacesXs are locally compact and there existsa finite setS0 S such thatXs is a compact fors S\ S0.Theorem. (3.3.15) If there exists an open mapping f: X Y of a locallycompact spaceX onto a Hausdorff spaceY, thenY is a locally compact space.

Theorem (The Whitehead theorem). (3.3.17) For every locally compact spaceX and any quotient mapping g : Y Z, the Cartesian product f = idXg : X Y X Zis a quotient mapping.

A topological spaceXis called a k-space ifXis a Hausdorff space and ifXis an image of a locally compact space under a quotient mapping.

Theorem. (3.3.18) A Hausdorff space X is a k-space if and only if for eachA X, the set A is closed in X provided that the intersection of A with anycompact subspaceZof the spaceX is closed inZ.

Corollary. (3.3.19) A Hausdorff space X is a k-space if and only if for eachA X, the set A is open in Xprovided that the intersection of A with anycompact subspaceZof the spaceX is open inZ.

Theorem. (3.3.20) Every sequential Hausdorff space - and, in particular, everyfirst-countable Hausdorff space - is a k-space.

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It turns out that local compactness of X is crucial; indeed, one can prove

(see Exercise 3.4.A) that if for a completely regular space X there exists anacceptable topology on the set RX then Xis locally compact.

Proposition. (3.4.4) For every family{Xs}sSof non-empty topological spacesand a topological spaceY, the combination :

sS(YXs) Y(

ss

Xs)is a home-

omorphism with respect to the compact-open topology on function spaces.

Proposition. (3.4.5) For every topological space X and a family{Ys}sS oftopological spaces, the diagonal :

sS

(YXs ) (sS

Ys)X is a homeomorphism

with respect to the compact-open topology on function spaces.

Lemma. (3.4.6) For every pairX, Yof topological spaces and every subbasePfor the spaceY, the setsM(C, U)whereCis a compact subset ofXandU P,form a subbase for the spaceYX with the compact-open topology.Theorem. (3.4.7) For every pairX, Zof Hausdorff spaces and every topolog-ical spaceY, the exponential mapping : Y(ZX) (YX)Z is a homeomorphicembedding with respect to the compact-open topology on function spaces.

Theorem. (3.4.8) For every topological space Y, a Hausdorff space Z and alocally compact space X, the exponential mapping : Y(ZX) (YX)Z is ahomeomorphism with respect to the compact-open topology on function spaces.

Theorem. (3.4.9) IfZ Xis a k-space, then for every topological spaceY theexponential mapping : Y(ZX) (YX)Z is a homeomorphism with respect tothe compact-open topology on function spaces.

Corollary. (3.4.10) IfX andZ are first-countable Hausdorff spaces, then forevery topological space Y the exponential mapping : Y(ZX) (YX)Z is ahomeomorphism with respect to the compact-open topology on function spaces.

Let Z(X) denote the family of all non-empty compact subsets of a HausdorffspaceXordered by inclusion (=).Z(X) is directed by . For any C1, C2Z(X) satisfyingC2 C1, and for an arbitrary topological space Y, a continuousmappingC1C2 : Y

C1 YC2 , viz.,C1C2 = i, wherei : C2 C1is the embedding;clearlyC1C2 (f) =f|C2 for any f YC1 .Theorem. (3.4.11) If X is a k-space, then for every topological space Y thespaceYX with the compact-open topology (with the topology of pointwise conver-

gence) is homeomorphic to the limit of the inverse systemS(X) = {YC, C1C2 , Z(X)}of the spaceYC with the compact-open topology (with the topology of pointwiseconvergence).

Lemma. (3.4.12) For every pairX, Yof topological spaces, any subsetA ofXand any closed subsetB ofY, the setM(A, B) is closed in the spaceYX withthe topology of pointwise convergence and, a fortiori, in the spaceYX with thecompact-open topology.

Theorem. (3.4.13) IfYis a regular space, the spaceYX with the compact-opentopology also is a regular space.

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Lemma. (3.4.14) Let X be a topological space and C a compact subspace of

X. Assigning to any f IX

the number (f) = supxCf(x) defines a function : IX Icontinuous with respect to the compact-open topology onIX .Theorem. (3.4.15) If Y is a Tychonoff space, then the space YX with thecompact-open topology also is a Tychonoff space.

Theorem. (3.4.16) If the weight of both X and Y is not larger then m 0andXis locally compact, then the weight of the spaceYX with the compact-opentopology is not larger thenm.

We say that a family Fof mappings ofXtoY isevenly continuousif for ev-eryx X, everyy Yand any neighbourhood V ofy there exists a neighbour-hoodUofx and a neighbourhoodW ofy such that [(F M({x}, W)) U] V, i.e., such that the conditions f F and f(x) W imply the inclusionf[U] V. It follows directly from the definition that if a family F of mappingsof X to Y is evenly continuous, then all members of F are continuous, i.e.,F YX .Lemma. (3.4.17) If Y is a regular space, then for every evenly continuousfamily of mappingsF YX the closureFof the setF in the Cartesian product

xXYx, where Yx = Y for every x X, is an evenly continuous family ofmappings, and, in particularF YX .Lemma. (3.4.18) IfF YX is an evenly continuous family of mapping thenthe restriction|FXof the evaluation mapping is continuous with respect tothe topology of pointwise convergence onF.

Lemma. (3.4.19) Let Y be a regular space, X an arbitrary topological spaceand YX the space of all continuous mappings of X to Y with the topology ofpointwise convergence. If a setF YX is compact and the restriction|F Xof the evaluation mapping is continuous, thenF is an evenly continuous familyof mappings.

Theorem (The Ascoli theorem). (3.4.20) IfXis a k-space andY is a regularspace, then a closed subset F of the space YX with the compact-open topologyis compact if and only ifFis an evenly continuous family of mappings and theset(F {x}) = {f(x) :f F} Y has a compact closure for everyx X.

The following theorem is a variant of the Ascoli theorem; the symbol F|Zthat appears in it denotes, for F

YX and Z

X, the family of restrictions

{f|Z :f F} YX .Theorem. (3.4.21) If X is a k-space and Y is a regular space, then a closedsubsetFof the spaceYX with the compact-open topology is compact if and onlyif F|Zis an evenly continuous family of mappings for every compact subspaceZ Xand the set(F {x}) ={f(x) : f F} Y has a compact closurefor everyx X.

3.4.A: If X is a regular space and there exists an acceptable topology onRX , then Xis locally compact.

3.4.E: A Hausdorff space X is hemicompactif in the family of all compactsubspaces ofX there exists a countable cofinal subfamily.

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(a) Prove that every first-countable hemicompact space is locally compact.

(b) Give an example of a countable hemicompact space which is not ak-space.

(c) Show that in the realm of second-countable spaces hemicompactness isequivalent to local compactness.

(d) Prove that if the space RX with the compact-open topology is first-countable and Xis a Tychonoff space, then X is hemicompact.

3.4.G (a) Show that nw(YX) w(X)w(Y) with respect both to the compact-open topology and to the topology of pointwise convergence on YX . Deducethat if X and Y are second-countable then YX is hereditarily separable withrespect both to the compact-open topology and to the topology of pointwiseconvergence.

3.5 Compactifications

LetXbe a topological space.A pair (Y, c), whereYis a compact space andc : X Y is a homeomorphic

embedding of X in Y such that c[X] = Y, is called a compactification of thespaceX.

Theorem. (3.5.1) X has compactification X is Tychonoff.Theorem. (3.5.2) Every Tychonoff space has a compactification Y such thatw(X) =w(Y).

We shall say that compactificationsc1Xandc2Xof a spaceXare equivalent

if there exists a homeomorphism f : c1X c2Xsuch that the diagramc1X

f c2X

X

c1

c2

is commutative, i.e., f c1(x) =c2(x) for every x X.Theorem. (3.5.3) For every compactification Y of a space X we have|Y| exp exp d(X) andw(Y) exp d(X).

Let c1X c2X if there exists a continuous mapping f: c1X c2X suchthatf c1= c2.Theorem. (3.5.4) Compactificationsc1X andc2Xof a spaceXare equivalentif and only ifc1X c2X andc2X c1X.Theorem. (3.5.5) Compactificationsc1X andc2Xof a spaceXare equivalentif and only if for every pairA, B of closed subsets ofXwe have

c1[A] c1[B] = if and only ifc2[A] c2[B] = . (16)Lemma. (3.5.6) Let A be a dense subspace of a Hausdorff space X and letf: X Y be a mapping ofX to an arbitrary spaceY. Iff|A : A f[A] Yis a homeomorphism, thenf[X\ A] f[A] = .

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Theorem. (3.5.7) If c1X and c2X are compactifications of a space X and a

mappingf: c1X c2X satisfies the conditionf c1= c2, thenf(c1(X)) =c2(X) andf(c1X\ c1(X)) =c2X\ c2(X).

Theorem. (3.5.8) For every Tychonoff space X the following conditions areequivalent:

(i) The spaceX is locally compact.

(ii) For every compactificationcXof the spaceX the remaindercX\ c[X] isclosed incX.

(iii) There exists a compactificationcXof the spaceXsuch that the remaindercX\ c[X] is closed incX.

The next theorem states an important property of the family C(X) of allcompactifications ofX.

Theorem. (3.5.9) Every non-empty subfamilyC0 C has a least upper boundwith respect to the order inC(X).Corollary. (3.5.10) For every Tychonoff spaceXthere exists a largest elementwith respect to the order inC(X).

The largest element inC(X) is called the Cech-Stone compactificationTheorem (The Alexandroff compactification theorem). (3.5.11) Every non-compact locally compact spaceXhas a compactificationX with one-point re-

mainder. This compactification is the smallest element inC(X) with respect tothe order, its weight is equal to the weight of the spaceX.Theorem. (3.5.12) If in the familyC(X) of all compactifications of a non-compact Tychonoff space X there exists an element cX which is the smallestwith respect to the order, thenX is locally compact andcX is equivalent tothe Alexandroff compactificationX ofX.

Theorem. (3.5.13) If a compact spaceYis a continuous image of the remain-der cX\ c[X] of a compactificationcX of a locally compact spaceX, then thespaceXhas a compactificationcX cXwith the remainder homeomorphic toY.

3.5.E Maximal compactification of a Tychonoff space Xcan be obtained by

taking the closure infF

If of the image of the space X under the mapping

fF

f, where F is the family of all continuous functions fromXtoIandIf =Iforf F.

3.6 The Cech-Stone compactification and the Wallman

extension

Let us recall that the largest element in the family C(X) of all compactificationsof a Tychonoff space X is called the Cech-Stone compactification ofXand isdenoted by X.

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Theorem. (3.6.1) Every continuous mappingf: X Zof a Tychonoff spaceXto a compact spaceZ is extendable to a mappingF: X Z.If every continuous mapping of a Tychonoff spaceXto a compact space iscontinuously extendable over a compactificationXofX, thenXis equivalentto the Cech-Stone compactification ofX.

Corollary. (3.6.2) Every pair of completely separated subsets of a TychonoffspaceXhas disjoint closures inX.

If a compactificationX ofXhas the property that every pair of completelyseparated subsets of the spaceX has disjoint closures inX, thenX is equiv-alent to the Cech-Stone compactification ofX.

Corollary. (3.6.3) Every continuousf: X I (Xis Tychonoff) is extendableto continuousF: X I.

If every continuous function from a Tychonoff spaceX to the closed inter-valI is continuously extendable over a compactificationX ofX, then X isequivalent to the Cech-Stone compactification ofX.

Corollary. (3.6.4) Every pair of disjoint closed subsets of a normal space Xhas disjoint closures inX.

If a compactificationXof a Tychonoff spaceXhas the property that everypair of disjoint closed subsets of the spaceXhas disjoint closures inX, thenXis equivalent to the Cech-Stone compactification ofX.

Corollary. (3.6.5) For every clopen subsetA of a Tychonoff spaceXthe closureA ofA inX is clopen.

Corollary. (3.6.6) For every compactificationY of a Tychonoff spaceY and

every continuous mapping f: X Y of a Tychonoff space X to the space Ythere exists an extensionF: X Y overX andY.Corollary. (3.6.7) If a subspace M of a Tychonoff space X has the propertythat every continuous function f: M I is continuously extendable over X,then the closureM ofM inX is a compactification ofM equivalent to M.If, moreover, M is dense inX, thenX=M.

Corollary. (3.6.8) For every closed subspaceMof a normal spaceXthe closureM ofM inX is a compactification ofM equivalent to M.

Corollary. (3.6.9) For every Tychonoff spaceXand a spaceT such thatXT Xwe haveT =X.

Theorem. (3.6.11) For every m 0 the Cech-Stone compactification of thespaceD(m) has cardinality22

m

and weight2m.

Corollary. (3.6.12) The spaceNhas cardinality2c and weightc.

Theorem. (3.6.13) For every pointxD(m) and each neighbourhood V ofx there exists an open-and-closed subsetU ofD(m) such thatx U V.Theorem. (3.6.14) Every infinite closed setF Ncontains a subset home-omorphic to N; in particularFhas cardinality2c and weightc.

Corollary. (3.6.15) The space N does not contain any subspace homeomor-phic to A(0), i.e., inNthere are no non-trivial convergent sequences.

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Corollary. (3.6.16) No non-discrete subspace ofN is a sequential space.

Corollary. (3.6.17) No space N {x} N, where x N\ N, is first-countable.

LetX be aT1-space and letD(X) denote the family of all closed subsets ofX. The family of all ultrafilters inD(X) will be denoted by F(X).

Properties of ultrafilters in F(X):

(1) / F.(2) IfA, B F, then A B inF.(3) IfB D(X) and B A = for every A F, then B F.(4) IfA F and A B D(X), then B F.

(5) IfA, B D(X) and A B F, then either A F or B F.(6) IfF = F, then there exist A F and A F such that A A = .

Ultrafilters that have an empty intersection are called free ultrafilters; theyform a subfamily F0(X) of the family F(X).

Letw X=X F0(X); for every open set U XdefineU =U {F F0(X) :A U for some A F } wX.

B= the family of all sets U whereU is an open subset ofX. The setwXwith the topology generated by the baseB is called the Wallman extension ofthe spaceX.

Theorem. (3.6.21) For every T1-space X the Wallman extension wX is a

quasi-compactT1-space that containsXas a dense subspace and has the prop-erty that every continuous mapping f: X Z of X to a compact space Z isextendable to a mappingF: w X Z.Theorem. (3.6.22) The Wallman extensionwX of aT1-spacesX is a Haus-dorff space if and only ifX is a normal space.

Corollary. (3.6.23) For every normal spaceXthe Wallman extensionwX isa compactification of the spaceXequivalent to the Cech-Stone compactificationofX.

3.7 Perfect mappings

A continuous mapping f: X

Y is perfect ifX is a Hausdorff space, f is aclosed mapping and all fibers f1(y) are compact subsets ofX. A one-to-onemapping f: XYdefined on a Hausdorff space Xis perfect if and only if itis a closed mapping, i.e., iffis a homeomorphic embedding and the setf[X] isclosed in Y.

Topological properties of Hausdorff spaces which are both invariants andinverse invariants of prefect mappings are called perfect properties; a class of allHausdorff spaces that have a fixed perfect property is called a perfect class ofspaces.

3.7.D: Let f: X Y be a hereditarily quotient mapping with compactfibers defined on a Hausdorff space X. Then w(Y) w(X) and ifX is locallycompact and Y is a Hausdorff space, then Yalso is locally compact.

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3.8 Lindelof spaces

We say that a topological space X is a Lindelof space, or has the Lindelofproperty,ifXis regular and every open cover ofXhas a countable refinement.

Theorem. (3.8.1) Every regular second-countable space is Lindelof space.

Theorem. (3.8.2) Every Lindelof space is normal.

Theorem. (3.8.3) A regular space X has the Lindelof property if and only ifevery family of closed subsets ofXwhich has the countable intersection propertyhas non-empty intersection.

Theorem. (3.8.4) Every closed subspace of a Lindelof space is a Lindelof space.

Theorem. (3.8.5) If a subspace A of a topological space X has the Lindelofproperty, then for every family

{Us

}sS of open subsets of X such that A

sS

Us there exists a countable set{s1, s2, . . .} Ssuch thatA i=1

Usi .

Theorem. (3.8.6) If there exists a continuous mappingf: X Yof a LindelofspaceXonto a regular spaceY, thenY is a Lindelof space.

Every regular space which can be represented as a countable union of sub-spaces each of which has the Lindelof property itself has Lindelof property. Inparticular, every regular space which can be represented as a countable unionof compact subspaces (Hausdorff spaces with this property are called-compactspaces) has the Lindelof and is therefore normal. Lindelof spaces are hereditarywith respect to F-sets.

Theorem. (3.8.7) The sum

sSXs, whereXs= fors S, has the Lindelof

property if and only if all spacesXs have the Lindelof property and the setS iscountable.

Theorem. (3.8.8) Iff : X Y is a closed mapping defined on a regular spaceX and all fibers f1(y) have the Lindel of property, then for every subspaceZ Y that has the Lindelof property the inverse image f1(Z) also has theLindelof property.

Theorem. (3.8.9) The class of Lindelof spaces is perfect.

Corollary. (3.8.10) The Cartesian productX Yof a LindelofXand a com-pact spaceY is a Lindelof space.

Theorem. (3.8.11) Every open cover of a Lindelof space has a locally finiteopen refinement.

The smallest cardinal number m such that every open cover of a space Xhas an open refinement of cardinality mis called the Lindelof numberof thespaceXand is denoted by l(X).

Theorem. (3.8.12) For every topological spaceXwe have l(X) nw(X).Example. (3.8.13) Niemytzki plane = separable, not Lindelof

A(m) for m> 0 = Lindelof space, not separableSince every countable regular space has the Lindelof property, it follows from

3.3.24 that there exist Lindelof spaces which are not k-spaces.

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Example. (3.8.14, 15) Sorgenfrey line K is a Lindelof space. K K is not.3.8.C: Observe that every hemicompact space is -compact but not neces-

sarily vice versa. For a locally compact space X the following conditions areequivalent:

(1) The space Xhas the Lindelof property.

(2) The space X is hemicompact.

(3) The space X is -compact.

(4) There exists a sequence A1, A2, . . . of compact subspaces of the space X

such thatAi Int Ai+1 and X=i=1

Ai.

(5) The space Xis compact or (, X)

0.

3.8.A Observe that X is hereditary Lindelof space if and only if all opensubspaces ofXhave the Lindelof property.

Show that a Lindelof space Xis a hereditarily Lindelof space if and only ifX is perfectly normal.

3.8.D: Prove that ifX andY are second-countable spaces and Y is regular,then the space YX is hereditarily Lindelof with respect to both compact-opentopology and the topology of pointwise convergence.

3.9 Cech-complete spaces

Theorem. (3.9.1) For every Tychonoff space X the following conditions areequivalent:

(i) For every compactificationcXof the spaceXthe remaindercX\ c(X) isanF-set incX.

(ii) The remainderX \ (X) is anF-set inX.(iii) There exists a compactificationcXof the spaceXsuch that the remainder

cX\ c(X) is anF-set incX.A topological spaceXis Cech completeifXis a Tychonoff space and satisfies

condition (i), and hence all the conditions, in Theorem 3.9.1.We shall say that the diameter of a subset A of a topological space X is

less than a coverA ={As}sSof the space X, and we shall write (A)


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3.10 Countably compact, pseudocompact and sequentially

compact spacesA topological space X is called a countably compact space if X is Hausdorffspace and every countable open cover ofXhas a finite subcover.

Theorem. (3.10.1) A topological space is compact if and only if it is a countablycompact space with the Lindelof property.

Theorem. (3.10.2) For every Hausdorff spaceXthe following conditions areequivalent:

(i) The spaceXis countably compact.

(ii) Every countable family of closed subsets of X which has the finite inter-section property has non-empty intersection.

(iii) For every decreasing sequenceF1 F2 . . . of non-empty closed subsetsofX the intersection

i=1

Fi is non-empty.

Theorem. (3.10.3) For every Hausdorff spaceXthe following conditions areequivalent:

(i) The spaceXis countably compact.

(ii) Every locally finite family of non-empty subsets ofX is finite.

(iii) Every locally finite family of one-point subsets ofX is finite.

(iv) Every infinite subset ofXhas an accumulation point.

(v) Every countably infinite subset ofXhas an accumulation point.

Theorem. (3.10.4) Every closed subspace of a countably compact space is count-ably compact.

Theorem. (3.10.5) If there exists a continuous mappingf: X Y onto Haus-dorff spaceY, thenY is a countably compact space.

Theorem. (3.10.6) Every continuous real-valued function defined on a count-ably compact space is bounded and attains its bounds.

Theorem. (3.10.7) If X is a countably compact space and Y is a sequential

space, in particular, a first-countable space, then the projectionp : X Y Yis closed.Theorem. (3.10.8) The sum

sS

Xs, where Xs= for s S, is countablycompact if and only if all spaces Xs are countably compact and the set S isfinite.

Theorem. (3.10.9) Iff: X Y is a closed mapping defined on a Hausdorffspace X and all fibersf1(y) are countably compact, then for every countablecompact subspaceZ Y the inverse imagef1(Z) is countably compact.Theorem. (3.10.10) The class of countably compact spaces is perfect.

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Theorem. (3.10.13) The Cartesian productXYof a countably compact spaceXand a countably compactk-spaceY is countably compact.Corollary. (3.10.14) The Cartesian product X Y of a countably compactspaceXand a compact spaceY is countably compact.

Corollary. (3.10.15) The Cartesian product X Y of a countably compactspaceXand a countably compact sequential spaceY is countably compact.

A topological space X is called pseudocompact if X is a Tychonoff spaceand every real-valued continuous function defined on X is bounded. One canreadily check that the last condition is equivalent to the condition that everycontinuous real-valued function on Xattains its bounds.

Theorem. (3.10.20) Every countably compact Tychonoff space is pseudocom-

pact.

Theorem. (3.10.21) Every pseudocompact normal space is countably compact.

Theorem. (3.10.22) For every Tychonoff spaceX the following conditions areequivalent:

(i) The spaceX is pseudocompact.

(ii) Every locally finite family of non-empty open subsets ofXis finite.

(iii) Every locally finite open cover ofX consisting of non-empty sets is finite.

(iv) Every locally finite cover ofX has a finite subcover.

Theorem. (3.10.23) For every Tychonoff space the following conditions areequivalent:

(i) The spaceX is pseudocompact.

(ii) For every decreasing sequenceW1 W2 . . . of non-empty subsets ofXthe intersection

i=1

Wi if non-empty.

(iii) For every countable family{Vi}i=1 of open subsets ofXwhich has finiteintersection property the intersection

i=1

Vi is non-empty.

Theorem. (3.10.24) If there exists a continuous mapping f: X

Y of a

pseudocompact spaceX onto a Tychonoff spaceY, thenY is a pseudocompactspace.


Xs, whereXs= fors Sis pseudocompactif and only if all spacesXs are pseudocompact and the setS is finite.

Theorem. (3.10.26) The cartesian product X Y of a pseudocompact spaceXand a pseudocompactk-spaceY is pseudocompact.

Corollary. (3.10.27) The cartesian product X Y of a pseudocompact spaceX and compact spaceY is pseudocompact.

engelking topology notes

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