ON MAPS: CONTINUOUS, CLOSED, PERFECT, AND WITH CLOSED GRAPH

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 2 (1997) 405-408

405

ON MAPS: CONTINUOUS, CLOSED, PERFECT, AND WITH CLOSED GRAPH

G.L. GARGandASHA GOEL

Department

of Mathematics Punjabi University Patiala-147002,India

(Received January 11, 1994 and in revised form September 29, 1995)

ABSTRACT.

This

paper

givesrelationshipsbetween continuous

maps,

closed

maps,

perfect

maps,

and

maps

withclosedgraph^{in certain}classesof topological

spaces.

KEY WORDS AND PHRASES.

Continuous,closed,perfect,closedgraph,

B-W

compact,Frechet, fiber,Hausdorff, regular, compact, countably compact.

1992

AMS SUBJECT CLASSIFICATION

CODES.54C05,54C10,54D30.

1.

INTRODUCTION.

Throughout,by a

space

we shallmeana topologicalspace. Noseparation axioms are assumed and no

map

^isassumed to be continuousor onto unless mentionedexplicitly; cl(A)willdenote the closureof thesubset

A

inthe

space X. A space X

issaid tobe

Tlat

^its

^{subset _A}

^{ifeachpoint of}

^A

^{isclosed in}

^{X. X}

is said tobe

B-W Compact 1I.!1

^if

^every

^{infinitesubset of}

^X

^{hasatleastonelimitpoint.}

^A

^{point xin}

^X

^is

said tobea

cluster

^{point limi___t in the}terminologyof Thron 1])of a subset

A

of

X

if

every

neighbourhood ofxcontains an infinitenumber ofpointsofA.

X

is saidtobe aFrechet

space

if whenever x ecl(A),there is a

sequence

ofpointsin

A

convergingtox.A

map

f:X

Y

is said to be perfectifitis continuous, closed,and hascompactfibers

f-1

(y), y ey.

For

studyofperfect

maps,

see [2]and its references.

The prima_,’y

purpose

of this

paper

is to giverelationshipsbetween continuousmaps,closed

maps,

perfect

maps,

and

maps

withclosedgraph.

A

generalization and ananalogueof theorem5of Piotrowski andSzymanski

[3]

andanaloguesof theorem 1.1.17 andcorollary 1.1.18 ofHamlett andHerrington [4]

arealso obtained.

NOTE.

The definitionsofsubcontinuous andinversely subcontinuous

maps

can befound in Fuller

MAIN RESULTS.

THEOREM

[4]

.Let

f:X

Y

be continuous, where

Y

is Hausdorff. Then f has closedgraph.

THEOREM

2.Letf:X

Y

be closedwithclosed

(compact)

fibers,where

X

isregular (Hausdorff).

Thenfhasclosedgraph.

PROOF.

We

prove

onlytheparenthesispart; the otherpart,^{which can}also beprovedⁱⁿa simple mannerbyusingour proofof theparenthesis part,hasbeen

proved

byFuller[5,corollary 3.9]and by Hamlett and Herrington[4,theorem1.1.17] bydifferenttechniques.

Let

xeX,

yeY,

yCf(x).Then

xf-l(y),

which iscompact.Since

X

isHausdorff,there existdisjoint

open

sets

U

andV containing

406 G. L. GARG AND A. GOEL

x and

f-1

(y) respectively.Then f is closedimpliesthere existsan

open

setWcontaining

y

such that f- (W)cVandtherefore,

f(U)W=.

It followsthatfhasclosedgraph.

Combiningtheorems and2,we get thefollowing

THEOREM

3.

Let

f:X

Y

beperfect,where either

X

or

Y

isHausdorff. Then f has closedgraph.

Thefollowingtheorem4(theorem5),part (b)of which isageneralization(analogue)of theorem5 of Piotrowski andSzymanski[3], givessufficient conditions under which theconverseof theorem

(theorem2) holds.

THEOREM

4.

Let

f:X

Y

have closedgraph.Thenfis continuous if

any

oneofthefollowing conditions is satisfied.

(a)

Y

iscompact,

(b)

X

isFrechetand

Y

is

B-W

compact, (c) fis subcontinuous.

PROOF. We

givetheproofofpart (b) only; part (a)iswell known(corollary 2(b)of Piotrowski and Szymanski[3],andtheorem1.1.10 of [4]),whilepart (c)is theorem3.4 of Fuller [5]. Let

F

bea closed subset of

Y

and let xeclf-

l(F)-f- I(F).

Since

X

is aFrechet

space,

there existsa

sequence {Xn}

^of pointsinf- (F)such thatx_{n x. Since}fhasclosedgraph,the set

H

of valuesof the

sequence

f(x

n)

^is

an infinite subset of the

B-W

compactset

F

and

F

is

T

atH.Therefore,

H

hasa clusterpoint

yeF,

y f(x),andthe set

U=X-f-1

(y)isan

open

set containing x. Thenxn x impliesthere exists a positive integer no such that

xneU

for all n_>no.Again fhas closedgraphand the set

K={xn:n_>no}U{x}

is compact;itfollowsthatf(K)is closed, which isacontradiction since it is

easy

toseethatyeclf(K)-f(K).

Hence

fmustbe continuous.

THEOREM.

5.

Let

f:X

Y

haveclosedgraph.Thenfisclosed ifany oneof thefollowing conditions is satisfied.

(a)

X

iscompact,

(b)

X

iscountably compactand

Y

isFrechet, (c) fisinverselysubcontinuous.

PROOF. We

givetheproofofpart (b) only; part

(a)

iswellknown(corollary2(a) ofPiotrowski and Szymanski[3]),while part(c)istheorem 3.5 of Fuller[5].Let

F

be a closed subset of

X

and let yeclf(F)-f(F).Since

Y

is Frechet and

T

atf(X),there exists a

sequence

f(x

n)

}ofdistinct points convergingto

y

wherex

nF.Now

the set of valuesofthe

sequence

_xn is aninfinite subset of the countably compactset

F

andtherefore,ithasaclusterpoint

xeF,

y f(x).^Since

Y

is

T

atf(X),theset V

=Y-{

f(x) isan

open

set containingy.Thenf(x

n) y

impliesthere exists apositive integer_{no such} that

f(Xn)eV

for all n_>o.^Sincefhas closedgraphand theset

K ={f(Xn):n_.>no}U{y

^iscompact,it followsthatf-

I(K)

is closed, which is acontradiction since it is

easy

^toseethat xeclf-

l(K)-f- I(K).

Hence

fmustbe closed.

Combiningtheorems and5(theorems 2and4), weobtainthefollowing theorem 6 (theorem 7), givingarelationshipbetween continuous and closed

maps.

Theorem 6 includes theorem16.19of Thron

],whiletheorem7includesandgives analoguesofcorollary 1.1.18 ofHamlett andHerrington

[4].

THEOREM

6.

Let

f:X

Y

be continuous, where

Y

isHausdorffandoneof the conditions(a), (b), (c)intheorem 5issatisfied. Thenfisclosed.

The condition that

X

iscountably compactintheorems5(b)and 6(b)cannotbereplaced bythe weaker condition that

X

isB-W compact,as the followingexampleshows.

EXAMPLE.

Let X=N,thepositive integers,with a basefor atopology on

X

the family of all sets ofthe form

{2n-l,2n},neN,

and

Y={0,1,1/2

^l/n asasubspace ofthe real line. Themapf:XY, definedby

f(2n-1)=l/n-l=f(2n)

for>n.2andf(1)---0=f(2),isa continuoussurjectionwhich is not closed, although

X

is

B-W

compactand

Y

isFrechet,Hausdorff.

ON MAPS: CONTINUOUS, CLOSED, PERFECT, AND WITH CLOSED GRAPH 407

THEOREM

7.

Let

f:X

Y

beclosed with closed(compact)fibers, where

X

isregular (Hausdorff) andoneof the conditions(a),(b), (c)in theorem 4 is satisfied. Then f iscontinuous(perfect).

Combiningtheorems and4, we obtain thefollowingrelationship between continuous

maps

and

maps

^withclosedgraph.

THEOREM

8.

Let

f:X

-Y

be

any map,

where

Y

isHausdorffandoneof the conditions (a),(b), (c)of theorem 4 is satisfied. Thenfiscontinuous ifandonlyif it has closedgraph.

Combiningtheorems 2 and5,we obtain thefollowing relationshipbetween closed

maps

and

maps

withclosedgraph.

THEOREM

9.

Let

f:X

Y

be

any map

withclosed(compact)fibers, where

X

isregular (Hausdorff)andoneof the conditions(a), (b), (c)of theorem 5 is satisfied. Then f is closed if andonly if ithas closedgraph.

Combiningtheorems3,4and5, weobtain thefollowing relationshipbetweenperfect

maps

and

maps

withclosedgraph.

THEOREM

10.Let f:X

-Y

beany map^withcompactfibers, whereeither

X

isHausdorff or

Y

is Hausdorffandoneofconditions(a), (b), (c)of theorem 4 andone ofthe conditions(a), (b), (c)of theorem’5aresatisfied. Thenfisperfectif andonlyif ithasclosedgraph.

COROLLARY. Let

f:X

-Y

beabijectionandoneof the conditions(a), (b),(c)of theorem 4 and one of the conditions(a), (b), (c)of theorem 5 be satisfied.Then f has closedgraphifandonly ifitis a homeomorphismand both

X,Y

areHausdorff.

Combiningtheorems8,9,and 10 weobtainthefollowing

THEOREM

11.

Let

f:X

Y

be

any map

^withclosed(compact)fibers,where

X

isregular (Hausdorff),

Y

isHausdorff,andoneof the conditions(a), (b), (c)of theorem 4 andoneof the conditions(a), (b), (c)oftheorem5aresatisfied. Then the following conditions (i) to(iii) (i)to(iv)}

areequivalent.

(i) f is continuous.

(ii) fisclosed.

(iii) fhas closedgraph.

(iv) f isperfect.

REFERENCES

1.

THRON,

W.J.Tot)oloicalStructures,Holt,Rinehart andWinston, 1966.

2.

GARG,G.L.&

GOEL,A.Perfect

maps

incompact (countably compact)

spaces, In.J.Math

^And

Mth.

Sci.

(To appear).

3.

PIOTROWSKI,Z.&

SZYMANSKI, A.Closed graphtheorem:topological

approach,RCndic0nti

Del Circolo

Matematico.

DiPalermo Serie

II

37(1988), 88-99.

4.

HAMLETT, T.R. &

HERRINGTON,

L.L.

Theclosedgraph^andp-closed graph propertiesⁱⁿgeneral topology,

Amer. Math. Soc. Providence

Rhode Island,1981.

5. FULLER,R.V.Relationsamongcontinuous and various non- continuous functions, Pacific

J.

Math.

25(1968),495-509.

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