K AND KEYWORDS

VOL. 4 (1997) 621-626

COVARIANT

AND CONTRAVARIANT APPROACHES

TO

TOPOLOGY

JERZY DYDAK

Department

ofMathematics, UniversityofTennessee.

Knoxville,Tennessee37996

(Received April 22, 1997 and in revised form June 12, 1997)

ABSTtLCT. This paper is an exposition ofresults contained in

[2].

The purpose of

[2]

^is

to present a way of viewing ofbasic topologywhich unifies quite a few results and concepts previously seemednotrelated(quotientmaps, producttopology,subspacetopology, separation axioms,topologiesonfunction spaces,dimension,metrizability). Thebasicideaisthatinorder toinvestigateanunknown spaceX, one eithermapsknown spaces to Xormaps

X

toknown spaces.

KEYWORDS AND PHRASES:function spaces,quotientspaces,localcompactness, compact- opentopology

1991 AMS SUBJECT CLASSIFICATION CODES: ^54B15,54B17,54C35,54C20 0. INTRODUCTION

In [1]

^theauthor presentedcertainresultsof basictopology from the point ofviewof Extension Theory.

In [2]

^webroaden the approach of

[1].

Namely,^extensiontheorycanbeviewed aspart of thecontravariantapproach, anditmakessensetoponderits dual,thecovariantapproach.

Supposewehave aclass of known spaces

K,

andwearefacedwith anunknown spaceX.

We

maychooseone ofthe followingstrategies:

1.

(Covariant

approach)

X

will be investigated by considering maps

f ^{K X}

^{from known}

spacestoX.

2. (Contravariant approach)

X

will be investigatedby considering maps

f

^{:X Kfrom}

^X

^to

known spaces.

Typesetby

A.ATEX

The covariant approachiswidely used in the clasicalhomotopythco.ryand leadsto homo- topy/homology groups

(see [11]).

^The contravariant approach is the mainstay ofshape the- o.ry

(see [10]),

(’ohomologicaldimensiontheo.ry(see

[12]),

^andleadstocohomology/cohomotopy groups.

However,

inbasictopology the prevalent approach^isthat ofintrinsic

definitions/theorems

in termsof open

sets/covers.

Thepurposeof this paper is to translate the intrinsic approach of basictopology into co-

variant/contravariant

approachesinanefforttounifyvariousconceptswhichseemed unrelated upto now (quotientmaps,product topology, subspace topology,separation axioms,dimension, metrizability). Webelievethat itbrings better understanding and betterresults. Also, ital- lowsto integrate basictopology withcategorytheory at an earlierstage. Letus explainthis statement inthecaseoffunctionspaces.

One ofthe fundamental concepts in category theory, is that of adjoint functors

(see [6]),

and in algebra one createsthe tensor product .s a left adjoint to the Hom functor. Thus, one proves the Adjoint Associativity Theoremof"algebra

(see

^{Theorem 5.10in}

[5]

^onp.214)"

Homn(M

(R)n

N,P) Homn(M, Homn(N,P)). In

the category ofsets a left adjoint tothe

i-Iom

functoristhecartesianproduct functor.

In

the category of topological spaces the problem is reversed: the cartesian product functor is well-understood and one has difficulty defining theHornfunctor. The difficulty liesinchoosinganaturaltopologyonthefunction space

yx

ofall maps from

X

toY. Through varioustrials anderrorstopologies created the so-called compact-opentopologywhich works very wellonthe categoryofcompactlygenerated spaces

(also

knownask-spacesorKelleyspaces).

In [2],

^amorenatural approachwaschosen: Sincethe bijectionadj)f

"Homs,(X, Homsct(Y, Z))--, Homst(X

^x

Y, Z)

^isgivenby adjy(f)(x,y)=

f(x)(y), let usdeclare

f" ^X HomTov(Y, Z)

^{tobecontinuousif}adj.(f) iscontinuous. This leads tothe basic covarianttopology on function spaces. Thus, the goal of

[2]

is, from the beginning,to discusstopologiesonfunctionspacessothat the resultingHornfunctoris aright adjointto the cartesianproduct functor via the function adjy, i.e. weare aiming at adj).

HomTo(X, HomTo(Y,Z)) HomTo(X

^x

Y,Z)

to be a natural homeomorphism.

In

the terminology of

[6],

adj). isthe adiugant equivalence

(or

ad_iugant) and,one is ledtothe front adjunction

(the

resulting natural transformationfrom the identity functortothe composition Horn^ox and the rearadjunction

(the

resulting natural transformation from thecomposition xoHomtotheidentity

functor).

^{The front}adjunctioncorrespondstothefunction

13

^adj)"(id) X

(X

^x

Y)"

givenbyB(x)(y) (x, y)^andisclearlycontinuousbythe definition of the basic covariant topology. The rear adjunction eval

X

^{x yX y} is the well-known evaluation function:

eval(x, f) f(x).

Thus, the question ofHorn being right adjoint tothe cartesian product reduces tothe question of continuity of the evaluationfunction. Itturnsout that it isconnectedtothe Whitehead-Michael

[9]

characterizationoflocally compact spacesasthose spaces

Z

^for which

f

^x

idz

^{is a quotient map if}

f

^{is a quotient map. This} leads quicklyto establishing that, on the category oflocallycompact spaces, the functor Homdefined asthe functionspace equipped withthe basic covariant topology ^{is a}right adjoint tothe cartesian product functor. Sincemostquestionsregarding compactlygeneratedspacescanbereducedto thelocally compact case,^{one can}apply the functor k

Top

kTopfromthe topological spaces

to the k-spaces and obtain Horn on the category, kTop which is right adjoint to the product functor (inkTop). Obviously,thisparticular functorHommstbeyX _withthecompact-open topology(bytheuniqueness ofadjoint

functors)

but the approach of

[2]

^{ismuchbetterintegrated}

withthccategory theoryand, thcrehre, allows naturaland functorialproofs.

There is adual approach^to theone described above: given a way of assigning topologies onfunctionspaceswetryto findtopologiesonthe cartesianproductsothat oneobtainsaleft adjoint functorto

G(Z) MapTo,(Y, Z)

if

Y

isfixed. Therearetwo cases ofinterest;Top PC being thepointconvergencetopologyor

Top

CObeing the compact-opentopology. Itisshown in

[2]

^thatone canintroducethe PC-productXxPCYandthe CO-product

X

x coYsothat ifTop

{PC, CO},

^thenonehasanatural equivalence

adjv

MaPTop(X

XTop

Y, Z) MapTo(X, MapTop(Y, Z)

forall spaces

X, Y,

Z. Onemay say that those productsare topologicalanalogsofthe temor product inalgebra. It turns out that thePC-product is commutativeandtheCO-product is not commutative.

In

the nexttwosections wediscussthebasicresults of

[2]

^withoutproofs. Thedetailedproofs canbe foundin

[2]

^{aswellas anextended}discussion of varioustopologiesonthefunctionspaces.

1. COVARIANT AND CONTRAVARIANTTOPOLOGIES

1.1. Definition. The topology

T

onX iscalled thecovarianttopolo__gv_ induced by the of functions:F

{ f ^X

I X

}

ifeach

X

_I isatopologicalspaceand

T

isthe largestof all topologiesonX undervchich all

.f ^:F

are continuous.

Noticethat thecovarianttopology^existsandconsistsof allsetsUsuch that

f-l(U)

^isopen

for each

f

^E

-.

^Thisleadsto:

1.2. Example.

A

surjective function

X Y

is aquotientmap iff the topologyonYisthe covarianttopologyinducedby the singlefunction

{}.

1.3. Example. Given aset

{Xs},es

of topological spaces, theclassicaltopologyonthe disjoint union

H xs

^{isthecovarianttopologyinducedbyinclusionsit}

^Xt H ^{x,, s.}

ES ES

1.4. Example. Givenasimpficial comple

K,

the

ea

topology

IKI

^{istheccreariantopology}

c

by

a mmsoas _!1 IKI, ee sp

K

a I1 I1 ^qPP

iththe standardmetrictopology.

Oneofthebasicclasses of topological spacesareFchet spaces

(see

Section 1.6of 1.5. Proposition. Xisa

Ecet

space^i_ffitstopology^{is tb,ecovariant}topologyinducedby

The dualtothenotionifthecovarianttopologyisthecontravariattopology:

1.6. Definition. The topology

T

oX is caed the contrvariattopolo_,_ iduced by

te

classo?fimctions

{ X X

_X ff

eac X!

^is topological space ad

T

is thesmallest

a

topologiesonX traderhch

a

j

"

^{arecontmuoas.}

Noticethat the (;ontravariant topology exists and its sub-basis consists ofall sets

f-l(u),

whereUisopenin

Xf

^forsome

f

^E’.

1.7. Example. ^/f

A

isasubsetofatopological space

X,

then the subspace topologyon

A

is thecontravariant topologyinducedby theinclusioniA

A

X.

1.8. Example.

An

injectivemap

f

^{X Y isa}homeomorphic embeddingiffthe topology onXis thecontravarianttopologyinducedby

{f}.

1.9. Example. Theproduct topologyon the cartesian product

I-I x.

^{is the} contravariant topologyinduced byprojections

{Tr 1-I ^{x, xt }tes.}

The basicpropertyofcovarianttopologiesis:

1.10. Proposition.

Suppose

thetopologyof

X

isthecovarianttopologyinducedbyaclassof functions

{f X

^{X}iej. Then, a functiong}

^X Y

^iscontinuousiffg^of, iscontinuous for a//zEJ.

Thebasicresultregardingcovarianttopologiesis:

1’.11. Theorem. Supposethetopologyon

X

isthecovariant topologyinduced by^a classof maps

{f, ^X, X},es

^{so thatX}

(J f,(X,).

If

Z

is locally compact, then thecovariant topologyinducedby

{f, idz

^{X, Z X}

Z},Es

istheproduct topologyonX

Z,

where each

X, Z

isequippedwiththeproduct topology.

The basicpropertyofcontravarianttopologiesis:

1.12. Proposition. Suppose the topologyofX isthecontravariant topology induced bya class of functions

{

f, X X,}ij. Then, a functiong Y X is continuousifff,^og is continuousforeach J.

2. BASICCONCEPTS IN TOPOLOGY

FROM COVARIANT/CONTRAVARIANT

POINTS OF VIEW

Let us assumethat thefollowingspacesarewell-understood:

1. Anti-discretespaces(spaceswiththe smallesttopologypossible),

2. Discrete spaces (spaces with the largest topology possible), including the integers

Z

and natural numbers

N,

3. S

(the

0-dimensionalsphereorthe simplest diretespacewhichisnotanti-discrete), 4. Theunit interval

I

withthestandardtopology,

5. Thereal numbers

R

withthe standardtopology.

Q c R

arerationals.

It is well known that connected spaces X are precisely those, ^sothat all maps

f

^X

S^o areconstant. Thus, connectedness isa contravariant property. Onthe other hand, path connectedness isa covariantpropertyas

X

ispathconnected iffany map

f

^S

^X

^extends

over I. Let usanalyze basic conceptsoftopology from those twopoints ofview.

2.1. Proposition.

1. Xis

To

^{i/itany map}

f ^A

^Xfrom an anti-discrete spaceA toX isconstant.

2. Xis

T

^ifianynon-constantmap

f

^So

^X

^{is a}homeomorphic embedding.

3. Suppose

X

^is

T1.

^Then,

X

is

T2 (Hausdorff)

iffS isanabsolute neighborhood extensorof X withrespectto finitesubspaces.

4. Suppose

X

is

To.

^Then,

^X

^is

Ta1/2

(Tychonoff) iffthe topologyofX is thecontravariant topologyinduced byafamily offunctions

{ f

^X

I}

s.

5. Suppose

X

is

TI.

^{Then, Xis}

T4 (normal)

iffS^O is anabsolute neighborhoodextensorofX.

6. Suppose X is

T1.

Then,

X

is collectionwise normal iff all discretespacesD areabsolute neighborhood extensorsofX.

Thepurest contravariant approximationof compactnessispseudo-compactness

(see

3.10of Xiscalled pseudo-comxactifany map

f

^X

^P

^{fromXtorealsisbounded.}

The followingresult summarizeswell-knowncharacterizations of Hausdorff compact spaces intermswhichare contravariantinspirit:

2.2. Theorem.

Suppose X

isHausdorff. The followingconditions areequivalent:

1. X iscompact,

2. X isregular and any map

f ^{X Y}

^{from X toaHausdorffspace isclosed,}

3. Xisregular and

f ^X

^isclosedin

^Y

for any map

f

^from

^X

^{to a}Hausdorff space

Y,

4. X is regular and

f

^{is a} homeomorphic embedding for any injective map

f

^{from X to a}

HausdorffspaceY.

Thefollowing well-known result ofTamano

(see

Theorem5.1.38 in

[3])

can be interpreted thatparacompactnessisacontravariantproperty:

2.3. Theorem

(Tamano). X

E

T2

^is paracompact iff

X

^x C is normal for all compact Hausdorff spacesC.

Thefollowing metrizabilitycriterionproved by the authorin

[1]

^meansthat metrizabilityis acontravariantproperty:

2.4. Theorem.

X To

^is metrizable iffthe topology of

X

is the contravariant topology inducedbyasetof maps

{fs X I}ses

^{such that}

Theorem2.4 wasimprovedin

[1]

^asfollows:

2.5. Theorem.

X To

^ismetrizable il7thereisa set ofmaps

{f5

^X

I}es

^{such that}

and

{f-l(0, 1]}ses

^isabasisofX.

=

Theorem 2.5impliesthewell-knownmetrizability criteria, Kuratowski-Wojdystawski Theo- rem,andArens-EellsTheorem

(see [1]).

Completnessinthesenseof(echisa covariantproperty

(see

Theorem 3.9.1 of

[3]):

2.6. Proposition. Suppose

X

isa Tychonoff space. Then,

X

^iscompleteinthesense of(ech iffany map

f ^A

^Xfromasubset

^A

^{of a} Tychonoff space

Y

extendsover a

G

^subsetofY.

Beingak-spaceisacovariantproperty.

(see

Theorem 3.3.18 of

[3]):

2.7. Proposition. Xis ak-space

(Ms()

^{knownascompactly}generated)iffthetopologyofX

isthecovariant topologyinduced byafunctiorsof maps

{f: c., x}

fromh)callycompactspacestoX.

Thefollowing well-knownresult underscores theimportance ofTheorem1.11

(see

^Theorem 3.3.27of

[3]):

2.8. Corollary. H X isak-space and

Y

islocally compact, thenX x

Y

is ak-spacc.

Being of coveringdimension nisacontravariantproperty

(see [7]):

2.9. Theorem

(Hurewicz-Wallman). dim(X) <_

niffS

e AE(X).

Theorem2.9 explainswhythecovering dimension^isthemostwidely used of all theories of dimension.

In [1]

theauthor proved the following generalization ofTietze-U.rysohn Theorem andU.rysohn Lemma:

2.10. Theorem. Suppose Y {point} ^isaHausdorff space. Then, the followingconditions areequivalent:

1.

Cone(Y) e AE(X),

2.

(Cone(Y),Y) e AE(X),

3.

Y e AWE(X).

Traditionally, the

Cone(Y)

of

Y

is understoodasthequotient spaceYx

I/Y

^x

(0}.

^That

wouldmeanthatthetopologyof theconei.sintroduced in a covariant manner. Ifonewants to mapspaces to thecone, then^asseeninProposition1.13;it isbetterto introduce atopologyon the conein a contravariantmanner. Noticethat thereare twonaturalfunctions: theprojection p

Cone(X)

^Iandtheprojectionpx

Cone(X)-

^pt X. Thesetwofunctions define a contravariant topology onthe cone

Cone(Y)

whichis equivalenttotheone introduced in

[1].

Thus,forgeneralspaces Y,onehastwokindsofcones: thecovariant coneand thecontravariant cone. In the case ofa metric space

Y,

the covariant conemay not be metrizable but the contravariant cone ismetrizable

(use 2.4).

Theorem2.10dealswith

con’travariant

cones.

REFERENCES

[1] J.Dydak,Extension theory: Theinterfacebetween set-theoretlcand algebraic topology, Topology andits Appl.20(1996),^1-34.

[2] J.Dydak, Covarant and contravamantpoints of^mewintopologyuath apphcatlons tofunction ^spaces, preprint(1997).

[3] R.Engelking, General Topology, Berlin, 1989.

[4] S.T.Hu, Theoryof^retracs,Wayne StateUniversity Press, 1965.

[5] T.W.Hungerford,Algebra, Springer-Verlag,NewYork, 1974.

[6] P.J.Hilton andU.Stammbach, A CourseinHomologicalAlgebra,Springer-Verlag, 1971.

[7] W.HurewiczandH. Wallmaa,Dlmenon Theory,PrincetonUniversityPress,1941.

[8] S.MacLane,Categoriesforthe working mathematician, Springer-Vedag, 1972.

[9] ^E.Michael,Local compactness and Cartesian products of^quotientmaps and k-spaces, Ann. Inst.Fourier 18(1968),^281-286.

[10] S.Matdeid andJ. Segal,Shapetheory,North-Holland Publ.Co., Amsterdam, 1982.

[11] E.Spanier, Algebraictopology,McGraw-Hill,NewYork, 1966.

[12] J.J.Walsh, Dimension, cohomologcaldmension, and cell-likemappings,LectureNotesinMath. 870, 1981, pp. 105-118.