Weakly closed multifunctions and paracompactness

Internat. J. Math. & Math. Sci. VOL. 16 NO. 2 (1993) 345-350

345

WEAKLY

CLOSED MULTIFUNCTIONS AND

PARACOMPACTNESS

CAO JI-LING

Department

ofMathematics

Tianjin University Tianjin, 300072

People’sRepublic of China

(Received

onDecember 12, 1990 andinrevised formJune6,

1991)

ABSTRACT.

The purposeof the present paper is toinvestigate weakly closed multifunctions between topological spaces. We discuss basic properties of weakly closed multifunctions and

some results of Rose and Jankovi are extended.

Our

main theorems are concerning some

properties of paracompactness under weakly closed multifunctions. These theorems are

generalizations ofsomeresults duetoKovaeviand Singal.

KEY

WORDS AND PHRASES. Almost closed multifunction, weakly closed multifunction, paracompact, almost paracompact, nearly paracompact, -paracompact, a-paracompact, a-lll-paracompact.

1991AMS SUBJECT

CLASSIFICATION

CODES. 54C60,54D18.

1.

INTRODUCTION.

Singal and Singal

[1]

introduced the almost closedness for arbitrary functions between topological spaces. Almost closed functions were also called regular closed by some authorsin

literature, for example, Noiri

[2].

Another important generalization of closedness is the star

closedness by Noiri

[3]

which implies almost closedness. Recently,

Rose

and Jankovi:

[4]

introduced the weak closedness forfunctionswhich isamoregeneral concept. Basicproperties of

weakly closedfunctionswerealsoinvestigatedin

[4].

Roseand Jankovi6

[5]

discussed conditions

onweakly closed functions whichassure a Hausdorff range. These results are improvements of

Noiri’s results in

[2].

In

this paper, weak closedness for multifunctions is studied. Firstly, we

giveacharacterization of weakly closed multifunctions, thensomeconditions aregiven such that a weakly closed mutlifunctionhas aclosedgraph or stronglyclosedgraph, andrelated results of

[4,

5]

are extended. Finally, we investigate some properties of paracompactness under weakly closed multifunctions. Some results of

[6-8]

aregeneralized.

In

particular, it isshown that the image of an a-paracompact

(resp.

a-almost paracompact, a-lll-paracompact, a-almost

’l-paracompact)

subset under a weakly closed multifunction isalso a-paracompact (resp. a-almost

paracompact, a-lll-paracompact,a-almost

lll-paracompact)

undercertainconditions. 2.

DEFINITIONS

AND PRELIMINARIES.

Let (X,T) be a topoligical space. A C_X, intA and ctA denote the interior and closure ofA

regularopen

(closed)

ifA intclA(A clintA). The family of allregularopen subsets of(X,T)is

abase foratopology T*onX which iscalled the semiregularization ofT. AC_X is said tobestar closed if A is closed in

(X,T*).

A(X)is the family of all nonempty subsets of

x.

For

c_

t(X),

#

{U:U }, and

I1

denotes the cardinality of

.

Recall thataspace

x

is said tobeparacompctif every opencovering of

x

hasalocallyfinite

open coveringrefinement. X is said tobe -paracompact if every opencovering of cardinality < hasalocallyfiniteopenrefinement.

DEFINITION 1.

([9])

A

subset A ofaspaceX is said tobea-parcompact iffforevery

X-open covering

cv.

ofA thereexistsanX-locallyfinite X opencovering

*"

ofA which refines

.

DEFINITION 2.

([3])

A

space

x

is nearlyparacompactiffeveryregularopen coverofXhas

alocallyfinite opencoveringrefinement.

DEFINITION3.

([8])

A

subset Aofaspace X is said tobe -nearlyparcompact iff every X-regularopencoverofAhasanX-openX-locallyfiniterefinementwhich covers A.

DEFINITION

4.

([6,

8])

A

spaceX is almost paracompact

(almost

-paracompact)iffevery open covering of

x

(every open covering of X of cardinality _<

)

has an open locally finite refinement whose union isdensein X.

DEFINITION 5.

([8, 10])

A

subset A ofa space X is said to be a-almost paracompact

(a-almost

t-paracompct)

iff for every X-opencover

(X-open

cover

o

and

I1

_<

)

of A there

xists

an X-locallyfinitefamily of X-open subsets

*"

whichrefines andis that the family of

X-closures ofmembers of

"

formsacoverofA.

LEMMA

1.

([10])

A

space X is almost paracompact if andonly if eachregularopencoverof

x

hasanopenlocallyfiniterefinement whoseunion isdensein X.

COROLLARY.

Paracompactness

implies near paracompactness; near paracompactness

implies almost paracompactness.

DEFINITION 6.

([11])

A

subset Aofa space X is said tobe a-nearlycompact iff every X-regularopen coverofAhasafinite subcover.

DEFINITION

7.

A

function ]:X-.Y is

(1)

almost closed ify(C) isclosedfor eachregularclosedset Cc_

x

see

1];

(2)

starclosed ify(C) isclosed foreverystar closedC

c_

X

[see 3];

(3)

weaklyclosed if elf(intC)

c_

I(C) foreveryclosedC

c_

x

[see

4,

5].

Clearlywehave hefollowingimplications: closed-starclosed-,almost_closedweakltclosed.

A

multifunction F:X--,Y is a mapping from X into t(Yl. If AC_X, BC_Y, let

F(A) LI{F(x):x

_

A}, F₊(A) {l

.

Y:t

6.F(x) for some x A and F(x) if x A},

F-(B)={xa.X:F(x)taB#b},

and

F+(B)={x.X:F(x)_B}.

Recall that F:X--,Y is upper

semicontinuous at X

(u.s.c.

at

X)

if

F+n(V)_

(x) for every V (F(x)), and F is lower

semicontinuous at r X

(1.s.c.

at x

X)

if

F-l(V)

E(z) for eachopen set V with

v

rF(x)#

.

F is u.s.c.

(1.s.c.)

onXifit isu.s.c.

(1.s.c.)

ateachx X; andFis continuousifit isbothu.s.c,and

DEFINITION 8.

([12])

A

multifunction F:X-,Y is almost lower semicontinuous at

z

X

(a.l.s.c.

atx

X)

iffcIF-

(V)

r(x) for eachopen set V withF(x)t3V

:

.

F isa.l.s.c, onXifit is a.l.s.c,ateachz X. Wecanprovethefollowinglemma easily.

LEMMA

2. The following statementsareequivalent forany multifunction F:X-,Y.

WEAKLY CLOSED MULTIFUNCTIONS AND PARACOMPACTNESS 347

(2)

F-

1(V)

C_intcIF-

I(V)

for all openV

c_

Y.

(3)

F(clV)C_cIF(U) for allopen UC_X.

We say that a multifunction F:X--,Y is

*-open

iffor each open Uc_X, ciF(U)=cloF(U); F is

irreducible if F(C)#Yfor each proper closed subset Cof X; and Fhas aclosedgraphif itsgraph

G(F) {(x,y):xEX,yEF(t)}isclosedin XxY.

DEFINITION 9.

([13])

A

multifunction F: X-.Yis said tohaveastronglyclosed

graph

iff for each pair(z,y).

x

xY-G(F) thereareU E(z) and

v

E(y)such that (U xclV)faG(F)=

.

Similarly to Definition 7, a multifunction F:XY is said to be almost closed

(star closed)

if F(C)is closed for each regular closed

(star closed)

subsetC ofX. F isweakly closed if for each closedC

c_

X, cIF(intC) C_F(C).

Otherbasicconceptsand terminilogy abouttopologicalspacesarereferredtoEngelking

[14].

3. SOME BASIC

RESULTS.

Let X and Y be topological spaces.

Now

we shall give a useful characterization ofweakly

closed multifunctionswhich isageneralizationofaresultin

[4].

THEOREM

1. Thefollowingstatementsareequivalentfor any multifunctionF:XY.

(1)

F isweaklyclosed.

(2)

For each open Uc_

x

and any B

c_

Y with

F-t{B)c_

U, there is an open V

c_

Y suchthat

B,

c_

Vandf-

(V)

c_

clU.

(3)

For

each yeY and each open set U

c_

x

with F-(y)c_U, there is V E/(y) such that

F-

(y)

_

ctu.

PROOF.

(1)

implies

(2).

Let UC_Xbeopen and Bc_ with

F-(B)C_U.

SinceF is weakly closed, we have F(X) clF(X) and clF(int(X-U))C_F(X-U)C_F(X). Let V Y-ciF(int(X-U)),

then

v

is open and B-F(X)C_Y-F(X)C_V, so that we have B=(BOF(X))t(B-F(X)) C_ F_{+ (U)tV}C_ V and

F-I(V)

X- F+

(clF(int(X-U)))

C_ X- F+

(F(int(X-U)))

C_ X-int(X-U) C_clU.

(2)

implies

(3).

Itisobvious.

(3)

implies

(1).

Let C be any closed subset of X, B Y-F(C), then

F-()C_

X-Cfor each

yB. Hence there is a

Vu(y)

such that

F-(V)C_cI(X-C).

Then V=

{Vu:

yB}c_Y is

open, and F-

(V)

J{F-

l(Vy):y

B}C_cl(X-C), intC X-cl(X -C)C_ X F-

(V).

Thereforeit

followselF(intC) C_clF(X F-

(V))

C_Y VC_Y B F(C).

THEOREM

2. Ifaweaklyclosed multifunctionF:X--,Yis

a.l.s.c.,

thenFis almost closed.

PROOF.

Let C be a regular closed subset of

x.

By

the’ weak closedness of F, cIF(intC)C_F(C). SinceF isa.l.s.c., by Lemma 2,

F-(Y-clF(inC))

C_

intclF-(Y-clF(intC))

C_ intcl(X-intC)=X-C. ThuselF(in,C)=F(C). ThereforeF isalmost closed.

COROLLARY

([5]).

Ify:x-Y isa weakly closed and almost continuousfunction,

then/"

is almostclosed.

Now

we shall consider conditions under whicha weaklyclosed multifunction has a closed or

stronglyclosedgraph.

THEOREM

3. LetF:XYbeaweaklyclosed multifunction ofaspace

x

intoaspace Ysuch that

F(y)

isO-closedfor eachy EY. ThenFhasaclosed

graph.

PROOF. Similartotheproofof Theorem4of

[5].

COROLLARY

3.1.

([5])

Ify:x-Y isaweaklyclosedfunctionthat

)’-(y)

isO-closed for each

y Y,then

f

hasaclosedgraph.

PROOF.

Since

x

isregular, then

F-l(y)

isO-closed for each y6Y.

Hence,

byTheorem 3, F

has a closed graph. Now suppose F is not u.s.c, at some _{point z0 X, there} must exist V E(F(0) such that F(U) Vfor each U E(z0)- Therefore we can choose nets {x,,:a D}c_X and {v,,:a D}C_ Y- V such that x,--,z0 andv,, F(z,,) for each D.

By

the compactnessof Y,

{v,:a D} must have a cluster point v0 Y-V. SinceG(F) is closed, then Vo F(zo). This is a contradiction.

THEOREM 4. Let F:X-.Ybe a

"-open

multifunction ofa space X intoa space Y. Then F hasastronglyclosedgraphif andonlyif it hasaclosedgraph.

PROOF.

Necessityisobvious. Sufficiencyissimilar totheproofofTheorem 3 of

[5].

COROLLARY

4.1.

([5])

Let f:x-Y be a "-open function. Then /" has a strongly closed

graphif andonlyif ithasaclosedgraph.

COROLLARY 4.2. Let F:X--*Y be a

*-open

and weakly closed multifunction such that

F-l(V)

is O-closed for eachv Y. ThenFhasastronglyclosedgraph.

4. MAIN THEOREMS

In

this section, we will investigatesome properties of paracompactness under weakly closed

multifunctions.

THEOREM 5. Let F:X-,Y bea weakly closed, u.s.c., and open multifunction ofa spaceX intoaspace Ysuch that

F-l(V)

isa-nearly compact for eachv6Yand F(z) isa-paracompactfor each 6X. Then F(K)isa-paracompactif K isana-paracompactsetofX.

PROOF. Let

{Uo:a

6 zX be any Y-open coverof F(K). SinceF(z) is a-paracompact for each 6K, then thereis aY-locally finitefamily of Y-open sets

*’x

whichcovers F(z)and refines

al.andsince F isu.s.c., {F +

l(x#

):(K}isanopencoverofK.

By

thea-paracompactnessof K, there is an X-locally finite and

x-open

cover

W’={wa:

v} of K which refines

{F+I(*’#):z6

K}. Consequently,for each V, thereis

za6

Ksuch that

F(Wa)

c_*Ca

#.

We

set

(;a

{F(Wa)r

V:V

C,a

for each t }, and(

o;

_{a. It} is not difficult to seethat ( isa Y-opencoverof F(K)which refines t. Nowweshall show that isY-locallyfinite. Foreachv Y,

since *, is X-locally finite, then there exists an

x-open

set

H,

containing z such that

clHx

intersects at most finite members of W" for each z

F-a(V).

Thus {intclHx:z

F-I(v)}

is an X-regular open cover of

F-I(V)

for each v Y. Since

F-(V)

is a-nearly compact, there exists {,z2, ,zn}C_F-(y) such that F-

(y)

c_ o(int

clHxi

C_int cl(O

Hx,)

_{H u.} Consequently,

Hu

is

X-open andintersects atmostfinitemembers of W.

By

the weak closednessof Fand Theorem thereexists

Qu

E(y) such thatF-(y) c_F-

a(Qu)

c_

cIHu

c_el(

Hi

). Itt’ollowsthat

Qu

must meet onlywith finitemembers of(. HenceF(K)isa-paracompact.

COROLLARY

5.1. If F:X-,Y is a weakly closed, u.s.c., and open multifunction of a

paracompact space X onto a space Y such that F-a(y) is a-nearly compact for each yeY, and F(z)isa-paracompact for eachz X, thenYisalso paracompact.

COROLLARY

5.2.

([7])

If F:X-Y is an almost closed, open and u.s.c, multifunction ofa

paracompactspace

x

ontoaspace Ysuch that

F-(y)

isa-nearlycompact for eachy Y andF(z)

is a-paracompact for each X, thenYisparacompact.

COROLLARY 5.3.

([8])

If

.:X-Y

is an almost closed, open and continuous function ofa

paracompact space X ontoaspace Ysuch that

f-(y)

is a-nearly compact for eachy Y, then Y isalso paracompact.

WEAKLY CLOSED MULTIFUNCTIONS AND PARACOMPACTNESS 349

THEOREM 6. Let F:X-,Y be aweakly

closed,

u.s.c., and open multifunction ofa space

x

intoaspace Ysuch that

F-(V)

isa-nearly compact for each vEY and F(z)is a-lll-paracompact

for eachz(;

x.

Then F(K)is a--paxacompactif K isana-’l-paracompact subset ofX.

COROLLARY.

If F:X-,Y is a weakly

closed,

u.s.c, and open multifunction of an

paracompact space X onto a space Y such that F(x) is a-:lll-paracompact for each X and

F-(V)

isa-nearly compact for eachv Y, thenYis-paracompact.

THEOREM 7. Let F:XY be a weakly

closed,

u.s.c., a.l.s.c, _{and open multifunction} ofa space X into a space Y such that

F-i(it)

is a-nearly compact for each

paracompact

(a-lll-paracompact)

for each qX. Then F(K) is a-almost paracompact

(a-almost

tll-paracompact)ifK isana-almost paracompact

(a-almost -paracompact)

subsetofX.

COROLLARY

([6]).

If .f:X--Yisaclosed,continuousandopenfunctionofanalmost -paracompact space X ontoaspace Ysuch that j’-

(v)

is compact for eachvEY, then Y isalso almost -paracompact.

THEOREM

8.

Let

F:X-Y be aweakly

closed,

continuousand irreducible multifunction ofa

space Xontoan almost paracompact space Ysuch that

F-(V)

is a-nearlyparacompact for each

it Yand F(z)is a-nearlycompact foreachz X. Then

x

isalmost paracompact.

PROOF. Let At _{U,,:a A be any regular open cover of X. Since F-I(y) is a-nearly

,paxacompact

for each It Y, there is an X-locallyfinite family of X-open sets

*’v

which covers

F-l(V

and refinesAt.

By

the weak closedness ofFand Theorem 1, there is

wv

EE(it)such that

F

(it) c_

F

I(Wv)

C_

cl(,’v

# for each ItqY. Thus weobtainan open cover W"

{wv:

_vq_{Y} of}Y.

Since Y is almost paracompact, there is a locally finite faanily ofopen subsets

refining such that

{cIDa:B

q V covers Y.

Hence

for each

B

E

,

there must be

v

Y such that

D

_C

Wu

and then F-

I(D)

C_F-

l(Wuo)

C_

cl(*’a

#). Since F is1.s.c., {F

(Da):

B

q V is a

family of open subsets of X, and then for each BV,

F-(Da)=F-(Da)t3cifua

#

=ci(F-(D,)taV’a#)t3F-(D/)C_cI(F-(Da)taf#).

Consequently, we have clF

-

(D)

Col(F_

-(D)f’l

#)

for each / V. Now let

(={F-(D)V:V_’f’u}

for each /V, and (=

(.

It is evidentthat ( isafamily ofopen subsets ofX whichrefines At. Firstly, weshall show that ( is locally finite.

To

do this,for each X, since F(z)is a-nearlycompact and is locally finite, thereis aregularopenG

c_

YwithF(z)C_ G andG meetsat most finitemembers of

.

Then F+

I(G)

intersects at mostfinitemembers of thefamily

{F-I(D):/

V andsince F is u.s.c.,F+

(G)

E(z). Thisshows that( islocallyfinite.

Now

weshallprovethat theclosures of members of form a covering of

x.

Since F is weakly

closed,

we’have

clDC_ciF(F-(D))

C_ cIF(int clF-

(D/))

C_F(cIF-

(D))

for each /E V. Therefore Y

OclD

C_ OF(ciF-

I(D))

F(UcIF-(D)).

SinceF isirreducibleand

{F-(D):/

6 V isclosure-preserving, being locally

finite, then

UcIF-(D)=

X and since each

(

v) is closure-preserving, being locallyfinite,

X

UciF-(D)

C_

Ucl(F-l(Do)’u

#)

C_

Ucl((/

# {clS:S (}

#.

Therefore,

by

Lemma

1,

x

isanalmost paracompactspace.

COROLLARY

([8]).

Let f:X--.Y bea star

closed,

continuous and irreducible function ofa space X ontoanalmostparacompact spaceYsuch that

y- (y)

is a-nearlycompact for eachIt Y.

Then X isalmostparacompact.

By

the similarproofofTheorem 8,we cangetthefollowingresults.

THEOREM 9. LetF:X--Ybeacontinuous, weakly closed andirreducible multifunctionofa space Xonto aspace Ysuch that

F-(y)

is a-’l-paracompact for each ItEY _{and F(z)} isa-nearly

COROLLARY

([6]).

If f:x--r is a closed,continuous and irreducible function ofaspace

x

ontoan almost -paracompact space Ysuch that

f-’(v)

is-compact for eachv Y. ThenX is

also almost -paracompact.

ACKNOWLEDGEMENT. The author would like to thank the referee for his valuable suggestions, Prof. D.A. Roseand Prof.D.S. Jankovifortheir kindof help.

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