A note on weakly quasi continuous functions

(1)

A

NOTE ON

WEAKLY

QUASI

CONTINUOUS FUNCTIONS

JINHAN PARK

Departmentof Natural Sciences PusanNationalUniversity ofTechnology Yongdang-dongNam-ku,

Pusan,

608-739,KOREA

HOEYOUNG HA

DepartmentofMathematics

InjeUniversity

Obang-dong,Kimhae, 621-749,KOREA

(Received September 27, 1994and in revised formFebruary25, 1995)

ABSTRACT. The notionofweaklyquasicontinuous functionsintroducedby

Popa

andStan ]. In this paper, the authors obtain the furtherproperties of such functions and introduce weak* quasi continuitywhich isweaker thansemicontinuity

[2]

butindependentof weakquasi continuity.

KEY WORDS AND PHRASES. Weakly continuous, semi continuous, weakly quasi continuous, weakly*quasicontinuous.

1991AMS SUBJECTCLASSIFICATIONCODE. 54C08.

1. INTRODUCTION

As weak forms of continuity intopological spaces, semi continuity, weak continuity [3], quasi continuity[4]and almostcontinuityinthe sense of Husain[5]arewell known. Neubrunnovi[6]showed thatsemicontinuityisequivalenttoquasi continuity. Also,Noiri [7]showed that semi continuity, weak continuity and almost continuityarerespectively independent. In 1973,

Popa

and Stan [1] introduced

weak quasi continuitywhich isimpliedbyboth weak a-continuity[8]andsemicontinuity. Itisshownin

[7]

that weak quasi continuity is equivalent toweak semi continuity due to

Arya

and Bhamini

[9].

Recently, Noiri in [7,8] investigated fundamental properties ofweaklyquasi continuous functions and comparedthe interrelationamong weak quasi continuity,weaka-continuity,semicontinuityand almost continuity.

Thepurposeof thispaperistoobtain some characterizationsof weakly quasicontinuousfunctions and investigatetherelationships between such functions andsomeseparationaxioms. Wealsointroduce

weak* quasi continuitywhich isweaker thansemicontinuity but independent of weak quasi continuity. 2. PRELIMINARIES

Throughoutthe presentpaper, spaces alwaysmeantopological spaces and

f

X

--,_Y_{denotes a}

singlevalued function ofaspace

X

into aspace Y. Let Xbeaspace and

A

asubset of

X.

Wedenote the closure ofAand the interiorof

A

by

CI(A)

and

Int(A),

respectively. A subset

A

is said tobe semiopen [2] (resp. preopen [10], -open

[11])

it"

ACI(Int(A))

(resp.

AInt(Cl()),

(2)

semiopensetis saidtobesemiclosed Theintersectionofall semiclosed setscontaining

A

iscalled the

semi-closure [13]of

A

andisdenoted by

s-Cl(A).

Thesemi-interior[13]of

A,

denotedby

s-Int(A),

is

definedbythe unionof all semiopen setscontained in

A.

A subset

A

of

X

is saidtobe regular open (resp regularclosed) [14]ifA

Int(Cl(A))

(resp A

Cl(Int(A))).

Apoint:rEXis inthe 0-closure of

A

[15], denoted by

Cl0(A),

ira t

CI(U)

#

0for each opensetUcontainingx. Asubset

A

iscalled 0-closedif

Cl0 (A)

A

DEFINITION A. Afunction

f

X

Y

is saidtobe

(a) semi continuous

[2]

(briefly, sc

iff-l(V)

E

SO(X)

for eachopensetVof

Y;

(b) almostcontinuous[5]iffor each:r

X

and each opensetVcontaining

f(x),

Cl(f-l(V))

is

aneighborhoodofz;

(c) weakly continuous [3] (resp. 0-continuous [16]) if for each :r X and each open set

V containing

f(x),

there exists an open set U containing x such that

f(U)C

CI(V)

(resp

f(CI(U))

C

CI(V));

(d) weaklya-continuous[8](briefly,

w.a.c.)

iffor eachz

X

and eachopenset Vcontaining

f(x),

thereexists a

U

a(X)

containingxsuchthat

f(U)

c

CI(V).

3. WEAKLY QUASI CONTINUOUS FUNCTIONS DEFINITION3.1.Afunction

f

X

--

Y

issaid tobe

(a)

weakly quasicontinuous (briefly, w.q.c.)iffor eachx E

X,

eachopenset

G

containingx

and each open set V containing

f(z),

there exists an open set U of

X

such that

0

#

UC G and

f(u) c o(v);

(b) weakly semi-continuous [9] (briefly,

w.s.c.)

if for each X and each open set V containing

f(x),

thereexists aU

SO(X)

containingxsuchthat

f(U)

CI(V).

Noidshowedin[7,Theorem4. thatafunction

f

X

Y

isw.q.c,ifandonlyiffor each X and eachopensetVcontaining

f(x),

thereexists aU

SO(X)

containing:rsuchthat

f(U)

c

CI(V).

Henceweknow thatw.q.c, andw.s.c,areequivalent concepts.

Thefollowingisshownin[7,Theorem 4.2,

4.3]

and

[8,

Lemma

5.3].

THEOREM3.2. Fora

function

f

X

Y,thefollowingareequivalent:

(a)

f

isw.q.c.

(b)

Foreach subset

B ofY,

s-Cl(f-l(Int(Cl(B))))

C

f-(Cl(B)).

(c)

Foreachregularclosedset

F ofY,

s-Cl(f-l(Int(F)))

C

f-l(F).

(d) Foreachopenset

B ofY,

s-CI(f-I(B))

C

f-I(CI(B)).

(e)

Foreachopenset

B ofY,

f-l(B)

C

s-Int(f-(Cl(B))).

09

Foreachregular ctoseasetB

of

Y,

f-1

(B)

SO(X).

(g) ForeachopensetB

ofY,

f-l(B)

C

Cl(lnt(f-l(Cl(B)))).

THEOREM3.3. Fora

function

f

X ---.Y,thefollowingareequivalent: (a)

f

isw.q.c.

(b) ForeachsubsetB

ofY,

-CI(f-I(B))

c

f-a(cIo(B)).

(c)

Foreachsubset

A

of

X,

f(s-Cl(A))

C

CIo(f(A)).

(d) Foreach subset

A

of

X,

f(Int(Cl(A)))

CIo(f(A)).

(e)

ForeachsubsetB

ofY,

Int(Cl(f-(B)))

c

f-(Clo(B)).

09

ForeachopensetB

of

Y,

Int(Cl(f-l

(B)

f-I

(CI(B)

).

PROOF. Itfollowsimmediatelyfrom Theorem 3.2 and 17, Theorem

1.5].

THEOREM 3.4. A

function

f

X Y is aw.q.c,

if

andonly

iffor

each subset B

of

Y,

s-Cl(f-l(lnt(Clo(B))))

C

f-l(Clo(B)).

PROOF. Necessity. Let B be a subset of Y. Assume that

x6 f-l(Clo(B)).

Then

(3)

implies that

CIo(B)

AW

0

and so W CY-

CIo(B),

e

CI(W) c CI(Y- CIo(B))

Since

f

is wqc, there exists a UE

SO(X)

containing z such that

f(U) c CI(W) c CI(Y- CIo(B))

This

implies that

Uf l(lnt(Cl0(B)))--0

and hence z

qs-Cl(f

(Int(Cl0(B))))

Therefore,

s-Cl(f

(lnt(Cl0(B))))C f I(CI0(B))

Sufficiency Let B be anopenset ofY Thenclearly

CI(B)

Cl0(B)

Byhypothesis, wehave

s-Cl(f

(Int(Cl(B))))=s-Cl(f-(Int(Clo(B))))c f (Cl0(B))-f (CI(B))

Hence, by Theorem 32,flswqc

The composition oftwo wqc functions may fail to be wqc

[7]

But Noiri showed m [7,

Theorem616]that under certainconditions the composition oftwofunctionsiswqc

THEOREM3.5. Let

f

X Yand9 Y Zbe

functions.

(a)

If f

isw.q.c, and tsO-continuous, then o

f

isw.q.c.

(’b)

If f

ISs.c.and isweaklycontinuous,then o

f

isw.q.c.

PROOF. (a) Let:r EXandWbe anopenset ofZcontaining

9(f(z))

Since9is0-continuous, there exists anopenset V ofYcontaining

f(:c)

such that

9(CI(V))

c

CI(W)

Since

f

is wqc, there

exists aU

SO(X)

containing:rsuch that

f(U)

C

Cl(V)

Hence

g(f(U))

C

9(Cl(V))

c

CI(W)

(b) Theproofiseasyand hence omitted

COROLLARY3.6(Noiri [7])

If f

X Ytsw.q.c, and g Y Z tscontinuous, then o

f

sw.q.c.

LEMblA 3.7 (Noiri and Ahmad [18]) Let A andB be subsets

of

X.

If

A

PO(X)

and

B

SO(X),

thenA B

SO(X).

THEOREM3.8.

If f

X

Ytsw.q.c, and

A

E

PO(X),

then the restnctton

flA

A

Yis

w.q.c.

PROOF. Let

:tEA

and V be an open set ofY containing

f(z)

Since

f

is wqc, there

exists a U

SO(X)

containing :r such that

f(U)c

CI(V)

Since

A

PO(X),

by Lemma 37 :c A U

SO(X)

and

(f[A)(A

U)

f(A

U)

c

f(U)

c

CI(V)

Hence

f[A

iswqc

COROLLARY3.9(Noiri [7])

If f

X Yisw.q.c,andAtsopenInX, then therestncnon

f]A

A Ysw.q.c.

COROLLARY 3.10 (Arya andBhamini

[9])

If f

X

Y isw.q.c, and

A

o(X),

then the restrlcnon

flA

A

Ysw.q.c.

Sufficient conditionforafunctiontobe wq c,when it isgiventobeso in somesubspace,isgiven

inthefollowing

THEOREM 3.11. Let

f

X Y be a

funcnon

and

{A,[i

I}

bea cover

of

X such that

A

SO(X)

for

each I.

If

f[A,

A

Yisw.q.c,

for

each I,then

f

sw.q.c.

PROOF. LetVbearegularclosedsetofY. Then

(f[A)-l(v)

SO(A,)

Since

A

SO(X),

by Theorem 2 4 of[19],

(f]A,)-i(V)

SO(X)

for each iI But

f-(V)=

U,I((f[A)--(V))

Then

f-l(v)

SO(X)

because theunionof semiopensetsissemiopen[2] Hence, by Theorem32

f

iswqc

COROLLARY 3.12. Let

f

X Ybea

function

and

{A[i

E

I}

beacover

of

Xsuchthat

A

a(X)

foreach

I.

If

flA A

Ytsw.q.c.foreach I,then

f

tsw.q.c.

COROLLARY3.13. Let

f

X Ybea

function

and

{A[i

I}

beacover

of

Xsuch that

A,

isopeninX

for

each I.

If

f[A,

A

Yisw.q.c,

for

each I, then

f

isw.q.c.

DEFINITION3.14. Let

A

beasubsetofX Afunction

f

X

A

iscalleda wqc retracnonif

f

iswqc and

flA

istheidentityfunction onA

(4)

PROOF.

Suppose

that

A

is not semiclosed Then there exists a x X such that x

s-CI(A)

A

Since

.f

is awqc retraction,

f(z)

z Bythe

T2

property ofX,there existdisjointopen sets

U

and V such that

zU

and

f(x)V

which implies

UACI(V)=O.

Let

WSO(X)

containing z. Then

U

n

W

SO(X)

and hence

(U

f3W)A

# O

because z

s-CI(A).

Let y

(U

f

W)

A.

Sincey

A,

wehave

f()

U WN

A

C

U

and hence

f(v)

6

CI(V)

This

implies that

f(W)

CI(V)

because

W.

This iscontraryto the fact that

.f

is wqc Hence

A

is semiclosed inX.

In

[8],

Noiri showed that if

Y

isT2,

fl

X

Y

is sc.,

f2

X

Y

iswa.c and

f

f2

on a

dense subset of

X,

then

.fl

f2

on

X

Similarly,wehave

THEOREM3.16. LetYbe

T

and

f

X Ybe almostcontinuous.

If f2

X Ytsw.q.c.

and

tf

fl

f

onadensesubsetD

of

X, then

f

onX.

PROOF. Similartotheproof of[8,Theorem410]by usingLemma3.7.

THEOREM3.17. Let

Y

be Urysohnand

fl

X

Y

bew.q.c.

If f2

X

Y

isw.a.c, and

if

f

f2

on adense subsetD

of

X,then

f

f2

onX. PROOF. Similartotheproofof[8,Theorem4.10]. 4. GRAPHS OF FUNCTIONS

The graph of a function

f

:X

Y,

denoted by

G(f),

is the subset

{(x,f(z))lz

X}

of the productspaceX Y. Noiri[20] showed that

if.f

X YisweaklycontinuousandYisT2, then the graph

G(f)

is closed. Using

"w.q.c."

and "semiclosed" instead of"weakly continuous" and "closed" respectively,weobtain the following.

THEOREM4.1.

If f

X

-

Y

isw.q.c, and

Y

isT2, then

for

each

(z,

y)

G(f),

thereexist

U

SO(X)

andopensetVin

X

suchthatz

U,

y V

andf(U)

Alnt(CI(V))

.

PROOF. Let

(z,

y)

G(f).

Then y

f(z).

Since

Y

isT2,there exist disjointopensetsV and Wsuch that y Vand

f(z)

W. Thisimpliesthat

Int(Cl(V))

A

CI((W)

.

Since

f

isw.q.c., there existsU

SO(X)

containingzsuch that

f(U)

c

CI(W).

Hence

f(U)

fq

Int(Cl(V))

.

COROLLARY4.2.

Iff

X

Y

isw.q.c,andYis

T,

thenthegraph

G(f

issemiclosed

PROOF. It follows fromTheorem 4.1.

THEOREM4.3.

If f

X

-

Y

isaw.q.c,andsisO-closedsubsetinX Y, then

pl(S

fG(f))

issemiclosedinX,where_Plisthe projection

of

X YontoX.

PROOF. Letx

8-C1(pl

(S

G(f))),

whereSis a0-closedsubsetof

X

Y.

Let

U

andVbeany opensetsof

X

and

Y

containingzand

f(x),

respectively. Since

f

isw.q.c., byTheorem 3.2 x

f-l(v)

C

8-Int(f-l(Cl(V))).

Since

U

A

-Int(f-l(Cl(V)))

SO(X)

containing x,

(U

f"l

8-Irlt(f-l(Cl(W))))

f’lpa(,_q’

AV(f))

:/(:

.

Letx0

(U

-Int(f-a(Cl(V))))

NpI(S

NG(f)).

This implies that

(xo, f(zo))

S and

f(zo)

CI(V).

Therefore,

(U

x

CI(V))

ASC

CI(U

x

V)

S and consequently,

(z, f(z))

CI0(S).

Since S is 0-closed,

(x, f(x))

Sf

G(f).

Hencez Pl

(S

N

G(f)).

Thisshowsthat Pl

(S

G(f))

issemiclosed in

X.

COROLLARY 4.4.

If

f

X

Y

has a O-closedgraph

G(f)

and y X

-

Y

isw.q.c., then

{z

XlY(z)

v(z)}

issemiclosed.

PROOF. Since

{z

xIf(z)=v(z)}

=pl(G(f)G(g))

and

G(f)

is a 0-closed subset of

X

x

Y,

itfollowsfrom Theorem 4.3 that

{x

X]f(z)

g(z)}

is semiclosed.

COROLLARY4.5.

If

f

X

Y

isO-continuous, y

X

--

Y

isw.q.c, and

Y

isUrysohn, then

{z

Xlf(z)

g(z)}issemiclosed

PROOF. ItfollowsfromTheorem 7of[21 andCorollary4.4.

DEFINITION 4.6. Let

f:X-

Y be a function. The graph

G(f)

is said to be strongly semiclosed iffor each

(z,

y)

X

Y-

G(f),

there exist U

SO(X)

and V

SO(Y)

such that

(5)

LEMMA

4.7.

If f

X Y hasa stronglysemtclosedgraph

G(f)

tf

and only t[

for

each

(x,l)EX Y

G(f)

there exist UE

SO(X)

and V

SO(Y)

such that z

U,

V and

f(U)

3

s-El(V)

0

PROOF. It follows fromDefinition 46

THEOREM 4.8.

If f

X YISw.q.c, andYts Urysohn, then

G(f)

tsstrongly,emtclo.wdm XxY.

PROOF. Since

s-CI(U)

C

CI(U)

for each subset U of

X,

it follows mmediately fromLemma

47

5. WEAK*QUASI CONTINUITY

DEFINITION 5.1. A function

f

X Yis weakly* quasi continuous(briefly, w* qc it"for

each opensetVofY,

f-(Fr(V))

issemiclosed inX,where

Fr(V)

denotes thefrontierofV Every c function is w* qc but the converse isnot true as the following Example 5 2 shows Moreover,Example5 2and 53show thatwqc andw* qc areindependent of each other

EXAMPLE

5.2. Let X

{a,b,c},

7-=

{qS,

X,{a}}

and a

{cb,

X,{a},{bc}}

Let

f

(X,-) (X, a) be theidentity function Then

f

isw* qc However,

f

isnot c and hencenot wqc

EXAMPLE 5.3. Let X

{a,b,c},

7-=

{,X,

{a}}

and a

{q6,

X,

{b}}

Let

f

(X, 7-)

(Xa)

bethe identityfunction Then

f

is wqc but

f

isnotw* qc

Thewqc functions arenotgenerally c [7] Thenext twotheorems giveconditionsunderwhich wqc and c functions areequivalent A spaceXis saidtobeextremallydisconnected if the closure of eachopenset isopeninX

THEOREM5.4. Let

f

X Ybea

function

andXbeextremallydisconnected. 7hen

f

IS&c.

If

andonly

if

f

isw.q.c,andw*.q.c.

PROOF. Thenecessityisclear

Sufficiency Let:r Xand Vbeanyopenset containing

f(:r)

Since

f

is wqc,thereexists a

U

SO(X)

f(U) c

CI(V)

But since

f

is w* qc,

f-l(Fr(V))

f-I(cI(V)-

V)

is semiclosed and hence by Proposition of [22] U-

f-l(Fr(V))

SO(X)

Further

f(:c)

Fr(V)

implies :c

f-a(Fr(V))

The proofwill be

complete if we show that

f(z)

f(U

f-l(Fr(V)))

C V Let V U

f-l(Fr(V))

Then

f(v)

El(V)

But

f-l(Fr(V))

andso

f(v)

Fr(V)

El(V)

Vwhichimpliesthat

f(u)

V InTheorem 5.4,wecannotdropthe assumptionthatXisextremallydisconnectedasExample5 5

shows

EXAMPLE 5.5. Let X

{a,b,c,d},

7-

{4,,X,

{b},

{c}, {b,c}, {a,b,c}, {b,c,d}}

and

a={qh, X,

{a},

{c}, {a,c},

{a,b,c}}

Let

f:

(X, 7-) (X,a)

be the identity function Then

f

is

wqc andw*qc butnotsc

A spaceX is saidtobe tim-compact [14]ifeach pointofX has a base ofneighborhoodswith compact frontiers

THEOREM5.6.

If f

X Ytsw.q.c,withthe closedgraph

G(f)

andYisrim-compact,then

f

lss.c.

PROOF. Letx

c

XandVbeanyopensetcontaining

f(x)

SinceYisrim-compact,there exists anopensetWof Ysuchthat

f(x)

W

c

Vand

Fr(W)

is compact

Bec,ause

f

iswqc,thereexistsa

U

SO(X)

F(U)

c

CI(W)

Let V

Fr(W)

Since

f(x)

W which is

disjoint from

Fr(W), (x,V)

-

G(f)

Then since

G(f)

isclosed, there existopensetsU and V such that:r C

Uv,

_V

Vv

and

f(Uv)

Vv 0 The collection

{Vv] v

Fr(W)}

is anopencoverof

Fr(W)

Since

Fr(W)

is compact, there exist a finite number of points Vl,V2,...,W in

Fr(W)

such that

(6)

f(Uo) c

f(,n=lUy,)

C

,n=lf(Uy,)

which is disjoint from tA

zn=lVy,

and hence disjoint from

Fr(W).

Thus

f(Uo)

N

Fr(W)

0.

However

f(Uo) c

f(U) c CI(W)

Therefore,

f(Uo)

C

CI(W)

Fr(W)

c W Hence

f

iss.c.

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