Contra continuous functions and strongly S closed spaces

CONTRA-CONTINUOUS

FUNCTIONS

AND

STRONGLY S-CLOSED SPACES

J. DONTCHEV DepartmentofMathematics

Universityof Helsinki

00014Helsinki10, FINLAND

(ReceivedJune 27, 1994)

ABSTRACT. In 1989 Ganster and Reilly [6] introduced and studied the notion of LC-continuous functions viatheconcept oflocallyclosedsets Inthispaperweconsider astronger form of LC-continuitycalled contra-continuity Wecallafunction

f

(X, 7.)

(Y, r)

contra-continuous if the

preimage of every openset isclosed. Aspace

(X,

7-)

iscalledstrongly S-closedif it hasafinitedense subsetorequivalentlyifeverycoverof

(X,

7-)

byclosedsetshasafinitesubcover. We provethat

contra-continuous images of strongly S-closed spaces are compact as well as that contra-continuous,

/5-continuous imagesof S-closed spaces arealso compact. Weshow thatevery stronglyS-closed space satisfiesFCCandhenceisnearlycompact.

KEY WORDS AND PIIRASES. StronglyS-closed, closedcover, contra-continuous, LC-continuous,

perfectlycontinuous,stronglycontinuous,FCC.

1991 AMS SUBJECT CLASSIFICATION CODE. Primary: 54C10, 54D20, Secondary: 54C08, 54G99.

1. INTRODUCTION.

Thefieldof themathematical science whichgoesunderthenameoftopologyis concerned with all questions directly or indirectly related to continuity. General Topologists have introduced and investigatedmanydifferentgeneralizationsof continuous functions. Oneof the mostsignificant ofthose notions isLC-continuity. Ganster and Reilly [6] defined a function

f:

(X, 7-)

(Y, or)

to be LC-continuous ifthe preimage ofeveryopensetislocallyclosed. Aset

A

C

(X,

7-)

iscalledlocallyclosed

[4]

FG[25])

if itcanberepresentedasthe intersectionofanopen andaclosedset. The importance of LC-continuityisthatithappenstobe the dualofnearcontinuity precontinuity)tocontinuity,i.e. a

function

f

(X,

"r)

(Y, r)

iscontinuousifandonlyif it isLC-continuousandnearlycontinuous

[7].

Due tothis theoremwe canobtaininterestingand useful variationsofresults in functionalanalysis, for exampletheorems concerning open mappingsandclosed graph theorem 17,18,20,27,28]

Inthispaperwepresenta newgeneralization of continuity calledcontra-continuity. Wedefinethis

class offunctionsbytherequirementthattheinverseimageof eachopensetinthecodomain isclosedin

the domain. Thisnotion is astronger form of LC-continuity This definitionenablesus to obtainthe following results(1) Contra-contlnuousimages

of

spaces havingadense

finite

subsetarecompact and

(2)Contra-continuous,/5-continuousimages

of

S-closedspacesarecompact.

the space propertyofhavinga finitedense subset isequivalentto thefollowing property Everyclosed

coverhasafinitesubcover Hencethis is astronger formaS-closedness and we callspaces havingthis

property strongly S-closed Thusrestatingourresultwehave Contra-continuousimagesofstrongly

S-closed spaces are compact Moreover we observe that contra-continuity is properly placed between

Levine’s strong continuity[11] and Gansterand Reilly’s LC-continuity

[6]

In fact it is even aweaker form ofNoiri’sperfect continuity 16] Adecomposition ofperfectcontinuityispresented by showing that a function

f"

(X,

7-)

(Y,

)

is perfectly continuous ifand only ifit is contra-continuous and

(nearly)continuous

An

improvement ofaresult achievedbySingal and Mathur

[22]

isgiven by showing

that continuous, contra-continuousimages of almost compact spaces quasi-H-closed QHC)are compact.

The notion of continuity was generalized in 1958 by Ptak [20] He defined a function

f

(X, r)

(Y, or)

tobenearly

connnuous

iftheinverseimage of everyopenset in

Y

isnearlyopen in

X.

A

subset

A

ofaspaceiscalled nearlyopen[6]if

A

C Int

-,

where

(as

wellaseverywherein this paper) Int Aand

A

denote theinterior andthe closure respectively of

A c (X, 7-).

Nearlyopensetsare

well-knownaspreopensets In 15],atopology7- has beenintroducedbydefiningitsopensets tobe the a-sets, that is the sets

A

c X

with

A c

Int Int

A

A

function

f"

(X, 7-)

(Y,

or)is

called

a-continuous 15]iftheinverseimageof everyopensetinYis an c-set inX. Sometimesa-sets arecalled a-open.

In1967Levine introducedthe notion ofsemi-opensets. Aset

A

C

(X, 7-)

iscalledsemi-open 12] if

A

CInt

A

A

functionf

(X, 7-)

(Y, o)

is calledsemi-contmuous 12] if the inverseimage ofevery opensetin

Y

is semi-open in

X.

Notethatafunction is a-continuous ifandonlyif it is semi-continuous andnearlycontinuous[21

Semi-preopensetswere definedbyAdrijevi6in 1986. Aset

A

C

(X,

7-)

iscalled semi-preopen if

A

CInt

A.

Semi-preopensetsaresometimes called

-open.

A

function

f"

(X, -)

(Y, o)

iscalled

fl-continuous

iftheinverseimage ofeveryopenset in

Y

issemi-preopen in

X.

/3-continuity wasrecently

investigated byPopaandNoiri 19] Everysemi-continuousand every nearlycontinuous functionis continuousbutnotvice versa.

In

Section 2 we study contra-continuous functions, while Section 3 is devoted to strong

S-closedness. InSection 4 weshow that every strongly S-closed space satisfiesFCC semi-irreducible)

and henceisnearlycompact.

2. CONTRA-CONTINUOUS FUNCTIONS

DEFINITION 1.

A

function

f"

(X,

7-)

(Y,

o)

iscalledcontra-continuousif

f-1

(U)

isclosed in

(X,

7-)

for each opensetUin

(Y, a).

Thefollowingcharacterizationofcontra-continuitycanbe obtainedby usingthe sametechniqueof

the similar result involving continuity.

TIEOREM2.1. Fora

function

f

(X, 7-)

-

(Y, )

thefollowing conditionsareequivalent:

(1)

f

is contra-continuous.

(2)

Foreachz E

X

andeach closedset Vm

Y

wtth

f

(:v)

V, thereexistsanopenset

U

in

X

suchthatz

U

and

f

(U)

CV.

(3)

Theinverseimage

of

eachclosedsetin

Y

isopeninX. Sinceclosedsets arelocally closed,thenwehave

TIEOREM2.2.

Every

contra-continuous

function

isLC-continuous.

REMARK 2.4. In factcontra-continuityand continuityareindependent notions. Examples2.3

above shows that continuitydoesnotimply contra-continuitywhilethe reverse isshowninthe following

example

EXAMPLE2.5. Acontra-continuousfunction need notbe continuous Let

X

{a,b}

be the Sierpinski space by setting 7.

{3,

{a}, X}

and

a={3,{b},X}

The identity functions

.f

(X, 7-)

(X, r)

is contra-continuousbut notcontinuous

In 1960 Levine [11] defined a function

f"

(X, 7.)-- (Y,a)

to be strongly continuous if

f

(A)

c

f

(A)

forevery subsetAofXorequivalentlyiftheinverseimageofeverysetinYisclopeninX

11,Corollary2]. Thuswehave

THEOREM2.6.

Every

stronglycontinuous

function

iscontra-contlnuous, l’q

In 1984 Noiri

[16]

introducedthe notion of perfect continuity between topological spaces. By

definitionafunction

f

(X,

7-)

(Y, or)

iscalledperfectlycontinuousiftheinverseof everyopensetin

Y

isclopeninX. Hence.

THEOREM2.7.

Every

perfectlycontinuous

function

iscontra-continuous, l-I Clearlythe following diagram holds andnoneof the implicationsis reversible"

Contra-continuous

Strongly Perfectly

LC-continuous Continuous )Continuous

Continuous

Itis interestingto notethat indiscrete andlocallyindiscretespaces everyopenset isclosed) canbe characterizedviaperfectlycontinuous functions. Forexampleit iseasilyseenthat aspace

(X,

7-)

is indiscrete if and only if every function

f’(X, 7-)- (Y,r)

is perfectly continuous and locally

indiscrete if and onlyifevery semi-continuousfunction

f"

(X,

7.)

(Y,

r)

is perfectlycontinuousor

equivalently ifand only ifthe identity function

f"

(X, 7-)

(X, 7.)

is perfectly continuous. In

[14]

locallyindiscretespacesarecalledpartitionspaces.

THEOREM2.8. Foraset

A

c

(X, 7-)

thefollowingconditions areequivalent:

(1)

A

isclopen.

(2)

Aisa-open andclosed

(3)

A

isnearlyopen and closed

PROOF.

(1)

=

(2)

and(2) (3)areobvious.

(3)

=

(1)

Since

A

isnearly open,then

A

CInt

A.

Since

A

isclosed,then

A

CInt

A

IntA

or

equivalently

A

isopenand henceclopen.

Asaconsequenceofthe above decomposition ofclopensets wehave the following decomposition of perfect continuity:

THEOREM2.9. Fora

functlon f

(X, 7-)

(Y, r)

thefollowingconditions areequivalent:

(1)

f

isperfectlycontinuous.

(2)

f

is continuousandcontra-continuous.

(3)

f

is a-continuousandcontra-continuous.

(4)

f

isnearlycontinuousandcontra-continuous.

nearly-continuous and LC-continuous Infact inthis decomposition LC-continuity can be replaced by

sub-LC-continuity

[6].

A space

(X,

7-)

isalmost compact quasi-H-closed QHC)ifeveryopen cover has a finite proximate subcover subfamily the closures ofwhosemembers coverX)

THEOREM 2.10. The image

of

an almost compact space under contra-continuous, nearly continuousmappingiscompact.

PROOF. Let

f

(X,

7-)

(Y,

a)

be contra-continuous andnearlycontinuousandlet Xbe almost compact. Let

(V,)zei

beanopencoverofY. Then

(f-l(V,))ze

is aclosed, nearly opencoverof

X

due

to our assumptions on

f

By

Theorem 2.8

(f-(V,)),s

is a clopen cover ofX. Since

X

is almost compact, then for some finite

J

C

I

wehave

X

to

,:f-l(V,)

to

,af-(V,).

Sincefis

onto, then

Y

tO,es

V,

orequivalently

Y

iscompact. I-I

COROLLARY2.11. [22,Theorem3

4]

The image

of

analmost compact(

AC)

space undera stronglycontinuousmappngiscompact.

PROOF.

Every

stronglycontinuous function isboth continuous and contra-continuous

(see

the diagram below Theorem27). l-!

EXAMPLE

2.12. The composition oftwo contra-continuous functions need not be contra-continuous. Let

X

{a,b}

be the Sierpinski space andset7-

{,

{a},X}

andr

{,

{b},X}.

The

identity functions

f’(X, 7-)

--

(X, )

and

f’(X, )

-

(X, 7-)

are both contra-continuous but their

composition9o

f.

(X, 7-)

(X, 7-)

isnotcontra-continuous.

Howeverthe following theorem holds. Theproofiseasyandhenceomitted.

THEOREM2.13. Let

f

(X, 7-)

--,

(Y,

r)

and9"

(Y, r)

--.

(Z,

u)

betwo

functions.

Then:

(i) 9o

f

iscontra-continuous,

if

giscontinuousand

f

is contra-continuous.

(ii)

9o

f

iscontra-continuous,

if

giscontra-continuousand

f

is continuous.

(iii)

9o

f

iscontra-continuous,

if

f

andgarecontinuousand

Y

is

locally

indiscrete. 1"i

3. COVERING SPACES WITH CLOSED SETS

In

thissectionwewillgiveacharacterization ofspaces, wherecovers

by

closedsubsetsadmit finite subcovers.

DEFINITION2. AspaceiscalledstronglyS-closedifevery closedcoverof

X

hasasubcover.

REMARK

3.1. Notethat thenotion ofstrong S-closedness is independent from compactness. The realline withthe cofinitetopologyis acompact space,which is notstrongly S-closed, whileagain the real line butthis timewithatopologyin which non-voidopensetsaretheonescontaining the origin

(insuchcasestheoriginiscalleda generic point

14])

isanexampleofanon-compact, stronglyS-closed

space. Thus a

T.

stronglyS-closed space neednotbe finite.

A

spaceis

T.

if singletons areopenor

closed. HoweveraTl-spaceisstronglyS-closedifandonlyif it is finite.

THEOREM3.2

Every

stronglyS-closedspace XisS-closed.

PROOF. Itisshownin [8, Theorem

3.2]

that

X

is S-closedifandonlyifeveryregularclosed coverofXhasa finitesubcover. Sinceevery regular closedsetisclosed the theoremisclear. I-!

Notethatthereverse inthe theorem aboveisnotalwaystrue. The realline(infact any infinite

set)

withthe cofinitetopologyisanexampleofanS-closedspace, which isnotstrongly S-closed, sincethe trivialsubsets are theonly regularclosedsets.

THEOREM3.3. Foraspace

X

thefollowingareequivalent:

(1)

X

isstronglyS-closed

(2)

X

hasa

finite

dense subset.

PROOF.

(1)

=

(2)

Since

{{x}

a:E

X}

is aclosedcoverof

X,

then for somefinite setSC

X

(2)

=

(1) By(2)forsome finitesetS

{xl,

...,xT}

c

XwehaveS X If

(A,),

is aclosed

coverofX, thenfor each index k nthere exists an index

z(k)

E Isuch that xk E

A,k

Thus

{Xk}

C

Az(k)

and henceX (_J

.=l(Xk

C U

__IA,,.!

Thus

(A,(1),...,A,(,r))

is a finitesubcover of

(A,),

and soXisstronglyS-closed

COROLLARY 3.4.

If

Xis stronglyS-closed, tken tke set i(X)

of

all isolatedpoints

of

XIs

fimte

The followingresult can beeasilyverified Itsproofisstraightfoward

THEOREM3.5

Strong

S-closednesstsopen keredttary. [2]

Recall that aspace

(X, 7-)

iscalledd-compact[10]ifeverycoverofXbydense subsets has a finite subcover

THEOREM3.6. Everyd-compact, stronglyS-closed spaceXts

fimte

PROOF. ByTheorem3 3XhasafinitedensesubsetS Since

{S

U

{x)

x EX

\ S}

is adense

coverof

X,

thenbyassumption

X

\

Sis finite ThusXis finitebeing theunionoftwofinitesets NextwegiveaconditionunderwhichS-closedness implies strong S-closedness. Recall thata set

A

c

Xiscalledasemi-generahzedclosedset sg-closedset) [2]ifsC1A

c

Owhenever

A

COand

O is semi-open Here sC1A is the semi-closure of

A,

i.e the intersection of all semi-closed sets containing

A

Asetis seml-closed if itscomplementissemi-open Notethat the following implications hold andnoneof themis reversible.

Aisclosed

=

A

is semi-closed

=

Aissg-closed

A complementofansg-closed setiscalled sg-open. Itis notdifficulttoseethat a setisregular

closedifandonlyif it isbothclosedandsg-open Thus the following result holds.

THEOREM3.7.

If

every closed subset

of

aspaceXissg-open, thenXisS-closed

if

andonly

tf

it isstronglyS-closed. [-1

THEOREM3.8. Contra-continuous images

of

strongly S-closed spacesarecompact.

PROOF. Let

f

(X, 7-)

(Y, or)

be stronglycontinuousandonto. AssumethatXisstrongly S-closed. Let

(V,)ei

beanopencoverofY Then

(f-(V)),e

is aclosedcoverofX,

sincefis

contra-continuous. Thus forsomefiniteJCIwehaveX

LI,jf-I(v).

Sincefis

onto,thenY orequivalently

Y

iscompact. !-i

REMARK3.9 Inthe theoremabove, contra-continuitycannotbe reducedtoLC-continuity,since

therearestronglyS-closed spaceswhich are not compact andsincethe identityfunction isalways

LC-continuous.

TttEOREM3.10. Contra-continuous,

/%conttnuous

tmages

of

S-closed spacesarecompact.

PROOF. Assumethat

f

(X, 7-)

(Y, r)

is contra-continuousand/3-continuousaswellasthat

XisS-closed. Let

(V,),I

be anopencoverofY. Then

(f-l(v)),6i

is a coverof Xsuchthat forevery

EI,

f-l(v)

is closed and /3-open due to assumption Thus for each I we have

Int

f-l(v)

c

f-l(v)

c

Int

f-a(V,)

orequivalently each

f-(V,)

isregular closedinX. ThussinceXis

S-closed forsome finiteJ

c

IwehaveX LI

6jf-l(v)

Thenclearly,

sincefis

onto,Y i.e.

Y

iscompact. I-I

In the notion of the above given proofand since regular closed sets are semi-open, then the followingresult is obvious.

THEOREM3.11.

Every

contra-contmuous,_/-contmuous

functton

tssemt-continuous.

In1968Singal and Singal[23]introducedthenotionof almost-continuity

By

definition afunction

f

(X, 7-)

(Y,

or)

iscalledalmost-continuous[23]iffor each:r Xandfor each neighborhood Vof

that

f

(X,

7-)

(Y,

or)

isalmost-continuous ifandonlyiftheinverseimageofevery regular openset in

Y

isopeninX. Usingthesametechniquelike inTheorem 3.8and Theorem 3.10 one canprove easilythe following result:

THEOREM3.12. Almost-continuous images

of

stronglyS-closedspacesareS-closed. !"1

Finallywenotethatin 1980

Hong [9]

introducedanother stronger form of S-closednesscalled

RS-compactness. Hedefinedaspace

(X, r)

tobe

RS-compact [9]

ifeverycoverof

X

by regularsemi-open

setshasafinitesubfamilysuch that the interiors of its memberscover

X. A

set

A

iscalled

regular

semi-open

[5]

if forsomeregular openset

U

wehave

U

c A c U.

Sinceregular opensetsareclearly regular semi-open, thenwehavethefollowing implication:

RS-compact

StronglyS-closed

S-closed

REMARK

3.13.

RS-compactness

and strong S-closednessaretotallyindependent notions.

An

infinitespacewiththe cofinitetopologyisanexample ofan

RS-compact

spacewhich isnot

strongly

S-closed, sincethespacehasnonon-trivialregular semi-opensets. Forthereverselet

X

bethesetofall

positive integers with the following topology: r

{,

{1}, {2}, {1, 2}, X}.

Note that

{1}

U

{{2,n}

:n

3,4,5,...}

is a regular semi-open cover of

X

which does not have evena finite subcover. But

X

isclearly strongly S-closed, since

{

1,

2}

is a finitedense subset of

X.

4.

STRONG S-CLOSEDNESS VERSUS

FCC

A

space

(X,

’)

is an

FCC-space

[25]

semMrreduciblespace

[14])

ifevery disjoint family of

non-voidopen subsets of

X

is finite orequivalentlyif

X

has onlyafiniteamountof regular opensets.

X

is irreducible ifeverytwonon-voidopensubsets of

X

intersect.

A

space

(X, 7-)

isnearlycompact

[22]

ifevery opencoverofXhas a finitesubfamilysuch that the interiorsofitsclosures cover

X

orequivalently ifeverycoverof

X

by regular opensets

X

hasafinite

subeover.

In

this section we will show that the following diagram holds and none of its implicationsis

reversible:

StronglyS-closed

Irreducible

S-closed

Fee

QHC

Nearlycompact

LEMMA

4.1. Forasubspace

Sofa

space

(X, 7)

thefollowingconditionsareequivalent:

(1)

SisFCC.

(2)

SisFCC.

PROOF.

(1)

(2) Let

(V)i

be a disjoint family of non-void open subsets of

S.

Then

(2)

=

(1) Let

(S

N

U,)ez

be adisjoint family ofnon-voidopen subsets ofS. Then

(S

N

U,),et

is adisjoint family ofnon-voidopen subsets ofSandby

(2) I

isfinite. ThusSisFCC.

COROLLARY4.2.

If

aspace

X

hasadenseFCC-subspace,then

X

is

FCC.

!"! TItEOREM4.3.

Every

stronglyS-closed space

(X, 7-)

tsFCC.

PROOF.

By

Theorem 3 3Xhasafinitedense subspace A SinceAisfinite, thenAisFCC

By

Corollary4.2XisFCC. I-!

TItEOREM4.4.

Every

irreduciblespace

(X,

7-)

is

FCC.

PROOF.

By

Theorem 14in

14] X

isirreducibleifandonlyiftheonly regular opensubsets of

X

arethetrivialones.

Straight from the definitionswehave:

TItEOREM4.5.

If

aspace

(X, 7-)

isFCC,then

X

isboth S-closed andnearlycompact

Finallywepointoutthat anexampleofan

FCC-space

which is notstronglyS-closedis an infinite cofinite space. Thus even an irreducible space neednot be stronglyS-closed. On the other hand a stronglyS-closed space neednotbeirreducible: a finite discretespacewith atleasttwopointsis aneasy

example

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