R continuous functions

VOL. 15 NO. (1992) 57-64

57

R-CONTINUOUS FUNCTIONS

CH. KONSTADILAKI-SAVVOPOULOUandD.

JANKOVI(

Department

of Mathematics and Statistics University ofAuckland

Auckland, NewZealand

(Received May

8, 1989 andin revisedformJuly 27,

1989)

ABSTRACT.

A

strong form of continuity of functions between topological spaces is introduced and studied. It is shown thatin many knownresults, especially closedgraph theorems, functions under considerationareR-continuous. Several resultsintheliteratureconcerningstrongcontinuity propertiesaregeneralized

and/or

improved.

KEY WORDS AND

PHRASES. R-continuity,continuity,closedgraph property,rimcompactness. 1980

AMS SUBJECT CLASSIFICATION CODE.

54C10, 54D10.

1.

INTRODUCTION.

The purpose of this paper is to introduce a strong form of continuity of functions between topological spaces and study its properties.

We

call this new class of functions, R-continuous functions, becauseit is closely related tothe class ofregular topological spacesandis instrumental in discussing preservation of regularity under mappings. Since R-continuous functions into

T

spaces have closed graphs it is not surprising that in many familiar closed

graph

theorems considered, functions are R-continuous. For instance, a well known result that closed graph

functions intocompactspaces arecontinuous is strengthened by showingthat they are actually

R-continuous. It is also observed that R-continuous functions are encountered in various standard settings.

Among

others, weshow that continuousfunctions fromlocally compact Hausdorffspaces into Hausdorff spaces are R-continuous. We also establish numerous basic results about

R-continuityaswellasobtainseveral improvements of known results. 2.

PRELIMINARIES.

By

f:

X

--

Y

is meant an arbitrary function between topological spaces. For a subset

A

ofa

space X,c$A, int

A,

and BdA denote the closure, interiorand boundary of

A

in

X,

respectively. Recall that

A

is regularopen if

A

int cA. The family ofregularopen sets in

X

is abasefora topology on

X,

denoted by rs.

For

A CX,

the 8-closure

(Velieko

[1])

of

A

in

X

is

cesA

{x

X:

c.U

A

,

for each open

U

C

X

with x

U}.

A

is 8-closed if

A

cesA

and 8-opensets in

X

arecomplements of 8-closed sets. Foraspace

(X,

r),

thespace

(X,

r0)

isthe set

X

with the topology

78

of8-opensets. Itis clear that _r0C₇₃Cr andthat a space isregularif and only if r _r#. Recall nowthat afunction

.f:X

Y

is strongly8-continuous

(Noiri

[2])

(resp.

8-continuous

(Fomin

[3]),

weakly continuous

(Levine

[4]))

if for each x

X

and each open set

Y

KONSTADILAKI-SAVVOPOULOU AND D. JANKOVIC

/(U)

C

cV).

Note that

f:(X,r)--, Y

is strongly 8-continuous if and only if

f:(X,

rt)--,

Y

is continuous. It is known that strong 8-continuity

>

continuity

>

8-continuity

>

weak continuityand thatnoneof theseimplicationsis reversible.

However,

if the domain ofacontinuous function is regular then thefunction isstrongly8-continuous andifthe range ofa weakly continuous function is regular then the function is strongly 8-continuous. If q is a filter base on a space

X

we denote byad q the adherenceofty, that is f3

{cg.F: F

E

if}

and by c$q_the

filter base

{cF:

F

E

q}. R

stands for the set of real numbers. 3. BASIC

PROPERTIES

OFR-CONTINUITY.

DEFINITION 3.1.

A

function

f:X--.

Y

is R-continuous at xq

X

if for each open set V containing

f(x),

thereexists anopen setUcontainingxsuch that

cef(U)

CV.

A

function jr:

X

--

Y

is R-continuousifitis R-continuousateachx

X. Note

thatafunction

f:

X

Y

is R-continuous if open sets

U

and Vin Definition 3.1 are replacedby a basic open set and a subbasic openset,respectively. Also, it isclear that R-continuous functionsare continuous.

Moreover,

R-continuity implies strong8-continuity, while the converse is not true ingeneral.

We

will first characterize R-continuity and then provide some examples to distinguish between continuity,strong9-continuity and R-continuity.

THEOREM3.1. Let

f:

X Ybeafunction. Then thefollowingareequivalent.

(a)

fis R-continuous.

(b)

Forevery filterbase on

X,

if converges tox

X

then

cf()

convergesto

f(x).

(c)

For

every x

X

and for every open set

V

containing

f(x)

there exists an open set of

U

containingxsuch that

c.f(c.U)

C

V.

(d)

Forevery x(

X

and for every closed set

F

in

Y

with

f(x)

_

F,

thereexistopensets

U

andV such that

xeU,

F

CVand

f(cU)

NV $.

(e)

Forevery x(

X

and foreveryclosed set

F

in

Y

with

)’(x)

(

F,

thereexistopen sets

U

and

V

such thatx(

U, F

CVand

f(U)f

V

.

PROOF. Weproveonly

(a)

===>

(b), (b)

===>

(c)

and

(e)

==>

(a)

since

(c)

===>

(d)

and

(d)

===>

(e)

areobvious.

(a)

===>

(b).

Let xE

X,

let 5 beafilterbaseon

X

converging toxand let Vbean open set in

Y

containing

f(x).

Since

f

isR-continuous, thereexistsanopenset

U

containingxsuch that

c)’(U)CV.

Since 5 converges to x there is an

F

such that

F

CU.

Hence

c..f(F)

C

c)’(U)

C

V

and so, the filter

c]()

converges

)’(x).

(b)

===>

(c).

Let

xX

and let

V

be an open set in

Y

containing

f(x).

The family

{U::U

is an open in

X

containing

x}

is a filter base converging to x.

By

hypothesis

cf(5)

{cf(U):

U

is anopen set in

X

containing

x}

converges to

f(x).

So,

thereexists anopen set

Uz

containing xsuch that

c..f(Ux)C

V. This impliesthat fis R-continuous andconsequently

I(v) c I(u).

So,

()

c

v.

(e) ===>

(a).

Let x

X

and let

V

be an open set in

Y

containing

f(x).

Then

Y-V

is closedin

Y

and

f(x)

Y-

V. Put

F

Y-

V.

By

hypothesis,thereexist opensets

U

and

W

such

that x

_

U,F

CW and

f(U)

OW and so,

f(U)

C

Y-

W.

Therefore,

c,f(U)

C

c,f(Y- W)

Y-

WC

Y

andhence

f

is R-continuous.

and

EXAMPLE

3.1. Let

X

X

U

X2,

where

{(0,0)}

U

{(O,+nl---m)"n

1,2 ;m

2,3,...}

X

X

₂

{(1,)"n

1,2,...}

U

{(1,nl----t-

nl-)

n 1,2,...; m 2,3

and

X

isconsidered as asubspaceof theplane.

By

identifyingthe points

(0,1

+

nl---m)

and

(1,

+

nl--m)

forn 1,2,..., andm 2, 3,..., weobtainanopenquotientmap ontoanon-regularHausdorffspace. The range of the surjectionisnotregular, sincethe closedset

{(1,nl-):

n

1,2,...}

and the point

(0,0)

cannot beseparatedbyopen sets. As aconsequenceofourTheorem 6.1we have that the range of anR-continuous opensurjection isregular. Therefore,the quotient mapis not R-continuous, while beingstrongly0-continuous.

It is clear that a continuous function into a regular space is R-continuous. This observation

slightly generalizes Theorem 8 of

Long

and Herrington

[fi].

Furthermore, the following

characterizationofregularspaces isreadilyestablished.

PROPOSITION3.1. Let

Y

beaspace. Then thefollowingareequivalent.

(a)

Y

isregular.

(b)

The identityfunction i:

Y

Y

isR-continuous(resp. strongly

0-continuous).

(c)

Forevery space

X

everycontinuousfunction from

X

into

Y

is R-continuous

(resp.

strongly

0-continuous).

To offer another sufficient condition for R-continuity we need the concept of subweak closedness offunctions.

A function/:X

Y

issaidtobe subweakly closed

(Jankovi

and

Rose

[71)

if there exists a base % for the topologyon

X

such that

cry(U)C

f(cC,U)

for every

U

.

The importance of this notion lies in the fact that the projections from theproduct of spaces onto the co-ordinatespacesaresubweaklyclosed.

PROPOSITION

3.2. If

f:X

Y

is strongly0-continuous and subweakly closed,then

f

is

R-continuous.

PROOF. Since

f

is subweakly closed, thereexists a base for the topologyon

X

such that

err(u)

C

f(cg,

U),

for every

U

e

.

The strong 0-continuity of

f

gives that for every x

X

and every open V containing

f(x)

there exists

U

such that

f(cC,U)C

V.

The result follows by combining the previoustwofacts.

Tosee that theconverse of the Proposition 3.2. does not hold, consider the identityfunction i:

(R,

r)

---,

(R, a),

whereristhe usualtopologyon

R

andaisthe indiscretetopologyofR.

THEOREM

3.2. If

fiX

Y

is continuous,

X

is locally compact Hausdorff, and

Y

is

Hausdorff,then

f

isR-continuous.

PROOF. Since

X

is regular and

f

is continuous,

f

is strongly 0-continuous. The local compactnessof

X

implies that thereisabase for thetopologyon

X

consistingof open setswith

compact closures. Let

U

.

Since

f

is continuous,

f(c,U)

is compact in

Y

and hence closed because

Y

is Hausdorff. Therefore,

f

is subweakly closed and so, by Proposition 3.2, jr is

R-continuous.

Noticethat thespace

X

in Example3.1 isnotlocallycompact.

KONSTADILAKI-SAVVOPOULOU AND D. JANKOVIC

[8]),

if foreveryopen set

U

in

X

and every x

.

U, c,{x}

C

U.

It is known that regular spacesare

R

₀whiletheconverseis not trueand thataspace is

T

ifandonlyifit is

R

₀and

T

_0.

PROPOSITION

3.3. If

f:

X

--

Y

isanR-continuoussurjection, then

Y

is

R

_0.

PROOF.

Let y(

Y

and let V be anopen set containingy. Thereexists an xE

X

such that V

f(x).

Since

f

is R-continuous, thereis anopenset

U

containingxsuch that

cef(V)c

v.

This implies

ce{v}

CVandhence

Y

is

R

_0.

The previous result enables us toconstructinasimplewayexamplesofstrongly 0-continuous andhencecontinuousfunctions whicharenot R-continuous. Thefollowing exampleisinthatsense instructive.

EXAMPLE

3.2. Let

"

bethe discrete topologyon

R

andr the left ray topology onR. The identityfunction

i:(R,.r)-- (R,a)

is strongly 0-continuousbut not R-continuous, since

(R,a)

is not

R

₀

Nextwestate several basic properties ofR-continuousfunctionswithoutproofssincetheproofs aresimilar tothoseinthecaseofcontinuous functions.

PROPOSITION

3.4.

(a)

If

.f:X

Y

is continuous

(resp. R-continuous)

and

g:Y--, Z

is R-continuous, then the compositiongo

f:

X

Z

isR-continuous.

(b)

If

.f:X--, Y

is R-continuous and

g:Y

Z

is continuous and

closed,

then the composition 9o

f:X

Z

is R-continuous.

(c)

Let

f:

X

Y

andg:

Y

---,

Z

_{be functions. If the compositiongof:X}-,

Z

is R-continuousand

f

is anopensurjection, theng is R-continuous.

(d)

If

f:X-, Y

is an R-continuous function and

A

C

X,

then the restriction

f/A:A-,

B

is

R-continuous, forany

B

with

f(A)

C

B

C

Y.

(e)

If

f:X

Y

is a function such that

fi

.f/Ui:Ui

_-’

Y

is R-continuous for every iI, where

{Ui:

iI}

is anopencoverof

X,

then

f

isR-continuous.

(f)

If each

Xi#d,

then the product function

II.f

:IIXi--,

IIY

is R-continuous iff each

f

i:

Xi

-

Yi

is R-continuous.

4.

R-CONTINUITY AND CLOSED

GRAPH PROPERTY.

Since the class of R-continuous functions into

T

spaces is properly contained in the class of functions with closed graphs, a natural question arises. Under what conditions are closed graph functions R-continuous?

In

answeringthis questionwe will strengthen someknown closed

graph

theorems.

But

first we will showthat afunction possessing a strongcontinuity property will also possess some strong form of closed graph property if its range satisfies a certain, usually weak, separationaxiom.

Recall thata function

f:

X

-

Y

has a0-closedgraph

(resp.

0-closed graphwith respect to

X)

(Hamlett

and Herrington

[9])

if for every

(x,y)q’ G(f)

(G(f)

denotes the graph of

f),

there exist open sets

U

and

V

containing z and V, respectively, such that

(cU

x

c.V)f3G(f)=

(resp.

(cgU

x

V)

n

G(f)

$),

or equivalently,

f(c.U)

t3

ceV

$ resp.

f(c.U)

t3

V

$).

Note that

f:

X

Y

hasa0-closedgraphifand onlyif

G(.f)

c.oG(f

).

Our

firstresult is avariationofthe well known and useful fact that continuous functions into Hausdorff spaces have closedgraphs

(actually,

0-closed

graphs

withrespecttothe

domains).

THEOREM

4.1. If

f:X

--,

Y

_is an R-continuous

(resp.

strongly

0-continuous)

functionand

Y

PROOF. The first part of the result follows from the facts that R-continuous functions into

T

spaces have closedgraphs and thatcontinuous functions withclosedgraphshave 8-closedgraphs

with respect to thedonains. Toshow the secondpart, let xE

X,

y E

Y

and y

f(x).

Since

Y

is Hansdorff, thereexist open setsVandWsuchthat

f(x)

V,

y

W

andVf3c,W

.

_{The strong} 8-continuity of

f

implies that there exists an open set

U

containing x such that f(c$U)C V. Therefore, f(c$U)f3c,W and

f

hasa8-closedgraph.

The parenthetical part of Theorem 4.1 improves Theorem 12 of (Long and Herrington

[6])

where it was shown that a strongly 8-continuous function into a Hausdorff space has a 8-closed graph with respect to the domain. We remark that another improvement of this result can be obtainedby replacing the hypothesis ofstrong8-continuitywith8-continuity.

Notethat thefunctionconsideredinExample3.1 hasa8-closedgraphbutis not R-continuous. Therefore, the converse of the first part of Theorem4.1 does not hold. Our next exampleshows that there is a function with a 8-closed graph into a Hausdorff space which is not strongly 8-continuous.

EXAMPLE

4.1. Let r be the usual topology of

R

and a be the topology

{U-A:Uer

and

A

C

{:n

1,2,...}}

on R. Clearly, rCa and

(R,a)is

an Urysohn space. The identity function i:

(R,

r)

(,

a)

hasa0-closedgraph, butisnot strongly0-continuous sincea

o

_a3 rand is not

continuous.

By

Theorem 4.1 and Proposition 3.3 it easily follows that R-continuous functions onto

T

_O

spaceshave0-closedgraphswithrespect to the domains.

Theproofof thenext propositionisstraight forward and henceisomitted.

PROPOSITION

4.1. If

f:X

Y

is an injection with a 8-closed graph with respect to

X,

then

X

isHausdorff.

Combining Proposition4.1 and thefirst partof Theorem4.1 we mayget asufficient condition for the domain of a function to be Hausdorff.

But,

it is not difficult to see that a strongly

continuous injectionintoa

T

_Ospacehasa Hausdorffdomain. This observationimproves Theorem 4of

[6]

and also implies the known fact that

T

_Oregularspacesare

T

_3.

Wenowturn ourattention tothe closedgraphtheorems. Thefollowingclosed graphtheorem was established independently by

Rose

[10]

and Noiri

[11]:

every closed graph weakly continuous function into a rim-compact space is continuous. This result was generalized in

(Jankovi

[12]).

We will show that a closed graph weakly continuous function into a rim-compact space has a strongcontinuityproperty, namely, it is R-continuous. Actuallyourresult states that closedgraph

continuousfunctions intorim-compact spacesareR-continuous.

As

an immediate consequencewe will obtain a strengthening of the well-known theorem that functions with closed graphs into compact spaces are continuous. Recall that a space is called rim-compact, if its topologyhas an openbasisof sets whoseboundariesarecompact.

THEOREM 4.2. Iff:X--,

Y

is a weakly continuous function with a closed graph and

Y

is rim-compact, then

f

is R-continuous.

PROOF.

In

ordertoshorten the proofwe assumethat

f

is continuous.

Let

qbeafilter base on

X

converging to x and let

V

be a basic open set such that

f(x)e

V and

Bd(V)

compact.

Then

c,f(F)N(’-l,z)#

for every

FE*,

since

c,f(F)

cc,V,

c,f(F)NBd(V)#d.

The compactness of Bd(V)implies there is a

yadc,f(*)fqBd(V).

This gives 1adf(i*)fqBd(V).

Since

r*

converges to x and

f

has a closedgraph, /=

f(x). So,

f(x)Bd(V).

This contradiction establishesthatc,f(q)convergesto

f(,r). By

Theorem2.2,

.[

is R-continuous.

In

[7]

it was observed that closed graph functions into compact spaces are strongly tg-continuous.

By

Theorem 4.2wehaveanimprovement ofthisresult.

COROLLARY4.1. If

f:X

Yis afunction withaclosed graph and

Y"

is compact,then

f

is R-continuous.

In

[t5],

Long

and Herrington established that a tt-continuousfunction

f:X

Y

witha9-closed graphwithrespect to

X

into the rim-compact space

Y

isstronglytg-continuous.

[Theorem 10]

and also, that a function

f:X

witha 9-closedgraph with respect to

X

into compact space

Y

is

strongly0-continuous

[Theorem

11].

Our Theorem 4.2 andCorollary4.1 considerably improve their results.

Combining Theorem 4.1. and Corollary 4.1, we obtain the following characterization of

R-continuousfunctionsinto

T

compactspaces.

COROLLARY

4.2.

Let

f:X

Y

bea functionand let

Y

bea

T

compact space. Then

f

is R-continuous ifandonlyif

f

hasaclosedgraph.

Theorem 9 of

[fi]

andourTheorem4.1 giveasimilar characterization ofstrong tg-continuity. COROLLARY 4.3. Let

f:X-, Y

be a function and let

Y

be a minimal Hausdorff space. Then

f

is stronglytg-continuousifandonlyif

f

hasa9-closedgraph.

Weclosethis sectionbypointingout thataclosed graphfunctionfromafirstcountablespace intoacountablycompact spaceisR-continuous. ThisresultstrengthensTheorem 1.1.11in

[9].

5.

FURTHER

PROPERTIES OF

R-CONTINUOUS

AND

STRONGLY

9-CONTINUOUS

FUNCTIONS.

Itisawell known fact thatif continuous functions

f

and9 which may, intoaHausdorff space, agreeon adensesubsetofthe domain, then theycoincide.

In

this sectionweobtain similarresults forR-continuousand strongly9-continuous functions.

THEOREM 5.1. If

f:X

Z

is a function with a 0-closed graph with respect to

X

and

9:Y

Z

is astrongly0-continuousfunction, then the set

{(z,V)

E

X

xY:

f(z)

()}

is0-closed.

PROOF.

Let

A

{(a:,y)

XxY:

f(:)

9()}

and let

(x,y)A.

Then

f(x)

#

9()

and

(z, tl(V))

c:.

(X xZ)-G(f).

Since

f

hasa 0-closedgraphwithrespectto

X,

thereexists open setsU and W containingzand

g(/),

respectively, suchthat

f(ceU)fl

W

.

Thestrong 0-continuity of g implies that there is an open set

V

containing t such that

9(cgV)

CW. Therefore

f(c,U)

t3

g(c,V)

.

Weclaimthat

(c,U

x

c,V)

t3

A

andhence

A

c,oA.

Suppose

that there is a(z,y)_(ce,

Uxc,V)flA.

A. Then

f(z)=g(v),f(z).f(ceU)

and

9(V)

E

g(c,V)

implying

f(c,U)

f3

9(c,V)

#

dp. Thiscontradictionestablishes theproof.

By

analyzing theproofofTheorem 5.1,it is easy toseethat there arenumerous variationsof Theorem 4.1. The most important one is certainly the following: if

f:X

-

Z

hasaclosed graph and

g:Y

Z

is continuous, then theset

{(z,V)5

X

xY:

f(z)= g(V)}

is closed. Other possibilities

axelefttothe reader.

We now cite some results that follow from the previously established results or may be obtained in similarmanner.

THEOREM 5.2. If

f:

X

Y

hasa0-closedgraphwithrespect to

X (resp.

0-closed

graph)

and

g:X

Y

isstrongly0-continuous(resp.

0-continuous)

then theset

{xeX: f(x)=

g(x)}

is0-closed. The second partof the nextcorollaryimprovesTheorem2of

[6].

COROLLARY 5.2. If

f:X--

Y

is R-continuous (strongly

0-continuous)

and

g:X

Y

is

strongly0-continuous

(0-continuous)

and

Y

is

T

(Hausdorff),

then

{x

EX:

f(x)= g(x)}

is0-closed.

COROLLARY

5.3. If f,y:X

Y

are R-continuous

(resp.

strongly

0-continuous)

functions whichagreeona0-dense subset

A

of

X (i.e.

ceaA

X)

and

Y

is

T (resp. Hausdorff),

then

f

g. 6. PRESERVATION OFREGULARITY.

As wehavementioned in the introduction oneof the reasonsfor introducing the concept of

R-continuityis its intrinsic relationshipwiththe regularity separationaxiom. Asobservedin Willard

[13],

verystrongconditions must beimposedon asurjectionso thatitpreserves regularity. Chaber

[14]

established that

T

₃ is preserved bycontinuous closedopen surjections. Weobtain this result by use of R-continuity. First, we need to recall that a function

f:X--, Y

is almost-open

[15]

if

f(U)

C int

c.f(U)for

every

(basic)

open set

U

in

X. Open

functions axe almost openwhile the converseisnottrueeveninthe presence of R-continuityasthefollowing exampleshows.

EXAMPLE

6.1. Let r be the usualtopologyon [I andtrbe the simpleextension ofrby the set of rationals

Q

i.e. tr=

{(UQ)uV: U,

Ver}.

The identity function

i:(X,a)-,(X,r)

is

R-continuousand almost open but not open.

THEOREM

6.1. If

f:X

-

Y

is anR-continuousalmost-opensurjection, then

Y

is

regular.

PROOF. Let g E

Y

and let V be anopen set containing V- There exists an x

X

suchthat V

f(z)

and since

f

is R-continuous there is an open set

U

containing x such that

c.f(U)C

V. The almost openness of

f

gives

f(U)C

int

c.f(U)

and consequently,

yf(U)C

WCceWC

V,

where W int

c.f(U). Hence,

Y

isregular.

COROLLARY

6.1. Regularity is preserved under continuous open subweakly closed surjections.

PROOF. Since

X

is regular and f:X--,

Y

is continuous,

f

is strongly 0-continuous.

By

Proposition3.2 itfollows that

f

isR-continuousand by Theorem 6.1 that

Y

isregular.

REMARK

6.1. We pointout that replacing the hypothesisopen in Corollary 6.1 with almost-open does not strengthen the result, since almost open subweakly closed functions from regular spacesareopen.

Moreover,

if

f:

X

Y

isasubweaklyclosedfunctionfromaregularspace

X

such that

f(U)C

int

c.f(c.U)

for every open

U

in

X,

then

f

is open. Let us agree to call afunction

f:X

--,

Y

_{almost weakly open if}

f(U)C

_int

c.f(c.U)

for every

(basic)

open set

U

in X. Clearly, almost-open

===>

almost-weakly-open and almost-weakly-open functions fromregular spaces are almost open. Also, the class ofalmost-openfunctions is properlycontained in the class of almost-weakly-open functions. Toseethis recallfirst that afunction

f:X

Y

isweakly-open

(Rose

[16])

if

f(U)C

int

f(ce.U)

for every

(basic)

open set

U

in X. It wasshown in

[16]

that weak openness and almost openness axe independent. Since weak openness obviously implies almost weak openness, we conclude that almost weak openness is a common generalization of both weak openness and almostopenness.

Now,

it is clear that replacing

"open"

in Corollary 6.1 with "weak

open"

also does not strengthen the result. The proofs of previous observations axe left to the reader.

COROLLARY

6.2. If

f:X

--,

Y

_is acontinuousopen closed surjection and

X

is

T3,

then

Y

is

64

PROOF. Notethat

T

ispreservedby closed surjections andapplyCorollary 6.1.

Weremark that Corollary 6.2 improves Theorem 14.6in

[13].

Also,it may be noted that the well-knownresult thateach factor ofanon-emptyregular

(T1)

product spaceisregular

(T1)

follows from Corollary 6.1

(since

the projectionsarecontinuous, open, subweaklyclosed surjections). That

the product of regular

(T1)

spaces is regular

(T1)

follows from Proposition 3.4

(f)

by setting

Xi

Yi,

letting

fi

be the identityfunction on

Xi,

and then using the equivalence of

(a)

and

(b)

from Proposition 3.1.

REFERENCES

1.

VELIKO,

N.V.,

H-ClosedTopological

Spaces,

Mat.$b0

(1966),

98-112. 2.

NOIRI,

T.,

On8-continuousFunctions,

J.

Korean

Math.

Soc.

16

(1980),

161-166. 3.

FOMIN,

S.,

Extensionsof TopologicalSpaces,

Ann.

of Math.44_

(1943),

471-480. 4.

LEVINE,

N.,

A

Decomposition of Continuityin Topological

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Amer.Math.

Monthly 68

(1961),

44-46.

5.

MACDONALD,

S.and

WILLARD, S.,

Domainsof

Paracompactness

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J.

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XXIV(6) (1972),

1079-1085.

6.

LONG,

P.E. and

HERRINGTON,

L.L.,

Strongly$-continuousFunctions,

J.Korean

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18

(1981),

21-28.

7.

JANKOVI,

D.S. and

ROSE, D.,

Weakly ClosedFunctions, submitted.

8.

DAVIS,

A.S.,

Indexed Systems of Neighborhoods for General Topological

Spaces,

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Math. Mo.nthly 68

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886-893.

9.

HAMLET,

T.R.

and

HERRINGTON,

L.L., Th....e

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ROSE,

D.A.,

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477-481. 11.

NOIRI,

T.,

Weak Continuity andClosed Graphs, Cas.

Pes

Mat__..

101

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379-382. 12.

JANKOVI,

D.S.,

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Internat.

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Math. & Math.

Sci.

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615-619. 13.

WILLARD, S., General

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14.

CHABER,

J.,

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Fund.

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LXXIV

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197-208. 15.

WILANSKY, A.,

Topics in Functional Analysis,

Lecture

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Springer-Verlag,Berlin, 1967.

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ROSE,

D.A.,

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Opeenness,

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