Paracompactness with respect to an ideal

Sci. VOL. 20 NO. 3 (1997) 433-442

PARACOMPACTNESS

WITH RESPECT

TO

AN

IDEAL

T. R.HAMLETT

Departmentof Mathematics East Central University

Ada,OklahomaUSA74820

DAVID ROSE "SoutheasternCollege of

the AssembliesofGod 1000LongfellowBlvd. Lakeland,FloridaUSA35801

DRAGAN JANKOVI(

Dept. ofMathematical Sciences

CameronUniversity

Lawton,OklahomaUSA73505

(Received May26,1995and in revisedform October26,1995)

ABSTRACT.

An

ideal on a setXis anonemptycollectionof subsets ofXclosed undertheoperations of subset andfinite union. Givena topologicalspace Xandan ideal2" of subsets of

X,

Xis defined tobe

2"-paracompactifeveryopencoverof the spaceadmitsalocallyfiniteopenrefinement which is a cover for all ofX exceptfor a set in2". Basic results areinvestigated, particularly withregard to the 2"-paracompactnessoftwoassociatedtopologiesgeneratedbysetsoftheformU whereUisopenand E2"and U

{UIU

isopenandU- AE2",for someopenset

A}.

Preservationof2"-paracompactness

by functions, subsets, andproductsisinvestigated. Importantspecial cases of2"-paracompact spacesare the usual paracompact spaces and the almost paracompact spaces of Singal and Arya ["On m-paracompact

spaces",

Math.Ann.,181(1969), 119-133]

KEY WORDS AND PHRASES: ideal, compact, paracompact, H-closed, quasi-H-closed, nowhere dense, meager, (continuous, almost continuous, open, closed, perfect)functions, regularclosed, open cover, refinement, locallyfinitefamily, r-boundary(--codense)ideal,compatible(r-local)ideal.

AMSSUBJECTCLASSIICATION: 54D18,54D30

I. INTRODUCTION

The concept ofparacompactness with respect to an ideal was imroduced by Zahid in [1] The concepts of almost paracompactness [2] ofSingal and Aryaand para-H-closedness ofZahid [1] are specialcases

An ideal on asetXisa nonempty collection of subsets of

X

closed under theoperationsof subset ("heredity") andfinite union("finiteadditivity"). An ideal closed under countableunions ("countable

additivity")is calledaa-ideal. Wedenoteatopological space(X,r)with an ideal2"defined onXby

(X,r,2"). Given aspace

(X,-)

andA _CX,wedenotebyInb(A)andCL(A)theinteriorand closure ofA,

respectively, with respectto

-.

Whenno ambiguityispresentwe writesimplyInt(A) and CI(A). If x X,wedenote theopen neighborhoodsystematxby-(x); e.,-(x) {U -[x

U}.

Weabbreviate "ifand onlyif" with"iff" Theconclusion or omissionofaproofisdesignated by thesymbol"t"l" H. BASIC RESULTS

Letusbeginwiththe followingdefinition.

DEFINITION [1] Aspace(X,7-,2")is saidtobe2"-paracompact,orparacompactwithrespectto.2-, iffevery opencover

F

ofXhas alocallyfiniteopenrefinement"f(notnecessarilyacove.r)such that

X-U 7 2" Acollection 7of subsets ofXsuch thatX L7 E2iscalledan2"-coverof X.

Singal and

Arya

[2]define aspace(X,-)tobealmostparacompactifeveryopencoverI"ofXhas a

Given aspace(X,7.),wedenoteby

N’(’)

the idealofnowhere densesubsetsof(X,7-) Thefollowing theorem establishes that almost paracompactness and para-H-closedness are special cases of paracompactness.

THEOREMILl.

(1)

A space (X,7-)isalmost paracompact iff(X,7.)isN’(7.)-paracompact

(2)

[1] A

T2

space(X,7.)ispara-H-closediff(X,T)is,Af(-)-paracompact. ["!

Thefollowingobvious result isstatedforthe sakeof completeness

THEOREM IL2. If(X,-,T) is

’-paracompact

and

7

is an ideal on X such that

Z

C_ ,7, then

(X,7.,)is

J-paracompact.

Givenaspace(X,-,7), thecollection/(2-,7.)

{U-

I:UE7.,IE

2-}

is a basisforatopology

-’(Z)

finerthan7. [3]. When noambiguityis present wedenote

B(2-,7.)

by/and

7"*(2)

by

-*.

If

v-*,

then wesay 2"is7--simple. Asufficient conditionfor

I

tobe simpleisthe following: forAC

X;

iffor every a Athere existsU -(a)suchthatUfA62-,thenA62-. If(X,7.,Z)satisfiesthiscondition,then issaidtobe compatiblewithrespectto2[4]orIissaid tobez-local,denoted 2- 7.. If(X,7-)is an infinite discretespace,thenthe idealoffinite sets is7"-simplebutnot7.-local. Itisknown that

N’(-)

inanyspace [5]. Itisalso known [Banach Category Theorem,

6]

that.Ad(7.) 7. inanyspace where

fl4(7.)denotes thea-idealofmeager

(or

firstcategory)subsets.

Given aspace(X,7.,2)andACX,wedenoteby

A*(2-,7-),

orsimply

A*

when noambiguityis present, the following:

A*

{x

E

XIUnA 6

2- for every U

7.(x)).

For AC(X,-), it is known that

A*(N’(7.),7.)

CI(Im(CI(A)))

[5], and

A*((7.),-)

isregular closed [5]. There is no known"closed

form" for

A*

( (7"),7") For further detailssee

[3].

A very useful fact about locallyfinitefamilies is that theyare closure preserving. The following theorem extendsthis result.

THEOREMIL3. Let (X,7.,Z)beaspace and let

{A{c

6

ZX

bealocallyfinitefamilyofsubsets of

X. Then

The simpleproofisomitted, r"l

In

T

spaces,A withrespecttothe idealoffinite sets isthederived setoperator, usually denoted by

A

.

HenceTheorem II.3 showsthat in

T

spacesthe derived setoperatordistributesacross arbitrary unions of locally finite families. Since for A C_ (X,r),

CI(A)

A({0},7.),

the well known closure preserving property offinite families is acorollarytothelasttheorem.

Givenaspace(X,-,Z),wesay$isr-boundary_

[8]

orr-codenseif

Z

n

7. (3},i.e.eachmember of

has empty--interior Inthenexttheoremweshow that theclassof almost paracompactspacescontains the class of2--paracompactspaceswhenthe ideal

Z

is7.-boundary.

THEOREMH.4. If2-is7.-boundary, and(X,7.)is-paracompactthen(X,7.)isalmost paracompact

PROOF. If

L

is any open coverof

X,

let

’

be alocally finite open refinement ofL/suchthat

X U?62-. Since2-is7.-boundary,

0

Int(A

t3

V)

X Ci(X

(X-

t.J

’))

X Cl(0

The following theoremsexaminethe preservation of2--paracompactnessamongthetopologies

-,

-*,

and(q(7-)),wherethislast topologyisdefinedbelow.

THEOREM H.S. Let (X,%Z) be a space. IfZ is --simple, --boundary, and

(X,-*)

is paracompact,then(X,-)is

Z-paracompact.

PROOF. Let

{U4{c

G / bea--open cover ofX ThenL/is a

7.’-open

coverofX and hence has a

7.’-locally

finite

-’-open

precise refinement

{V4-

I41V4

E7.,

I4

62-, and c6

&

such that

X

U (V4

14)

J6I. Without lossofgenerality, assume

I

V4

U4

so that

U4.n

V4-

I4.

We claim that {V c

Z

is 7.-locally finite. Indeed, for x E

X,

there exists

U-

7.’(x)(U

G7.(x),I Z) such that

CU-

I)n (v4 I4) 0 for c

6

{c,c2,..,c}. If

PARACOMPACTNESSWITH RESPECT TO ANIDEAL

(UVVo) (IUIo)-

0.

ThisimpliesUV

Vo--0

since otherwiseUV

Vo

is a nonemptyr-open subset of U

I,

which contradictstheassumptionthatZisr-boundary If {Uy_{V,[aE A},}thenl}is

7-locallyfinite since

{V[c

E A is--locallyfinite. Also,Visar-openrefinementof/Aand is an2,-cover

ofXsinceX U (Uo

n

Va)

X tJ

Cv’a

Io)= J. [:]

If(X,-r,T)is aspace, wedefine asetoperator

:

P(X)

7,where

7(X)

isthepowersetof

X,

as follows

[7]

ifAC_X,thenA) X (X-

A)*

J{U6

TIU-

AE

2-}

Notethat 2-is"r-local ifand onlyifA) A62foreachA_C X. If

B

is a basisfor7,then(B) {gB)]B6

B}

is a basisfora

topologycoarserthan7,denoted

((B)).

Furthermore,

(b(B))

((T))

((T’))

[7] AlSO,if2-is

7-local,

((T)/

((7(X)))

sinceforAC_

X,

b(A)= b(b(A))

Let (X,7-,2"), beaspace. We saythatZisweakly r-localif

A*

0implies A62 2"is called-_:

locallyfinite if the union of each"r-locallyfinitefamilycontained in

Z

belongstoZ.

LEMMAII.6[3]. Let(X,T,Z)be aspace. ThenZisr-localimpliesZisweakly T-local. E! Itisremarkedin

[3]

thataspace

(X,-)

iscountably compactifand only if theidealoffinite sets,Zf,is

weakly r-local,whereas w-locality of Zi,isequivalenttohereditary compactness of(X,7). Thereforethe implicationinLemmaII.6 isnotreversible. The followingexampleshows thatanidealcan be r-locally finiteandnotwealdyr-local.

EXAMPLE. LetX [0,f),wheref denotesthe firstuncountable ordinal,and letTdenote the usual

9rder

topology on X. Denoteby the ideal of countable subsets of X. Since(X,-) is countably compact, anylocallyfirfitefamily of nonemptysetsmustbe finite. Consequently,theunionof any locally finitefamilycontained in

Zo

belongsto

Z,

andhence isT-locallyfinite. Sinceevery pointinXhasa coumable

neighbor.hood,

A*

for everyA_CX. Inparticular,

X*

butX

t

,

andhence

Zt

isnot

weakly T-local.

THEOREMII.7. 2"isweakly v--local implies 2"isr-locallyfinite. ["]

THEOREMII.$. If(X,T,Tis

Z-paracompact,

and

Z

isT-locally finite, then 2"isweaklyr-local

PROOF. Let

A

0.

Foreveryx6

X,

thereexists

Ux

6T(x)with

Ux

NA62" {Uxlx6X}isan

opencoverofXand hence there exists aprecise locallyfiniteopenrefinement {Vx]x6X}which is an 2-cover of

X;

i.e., X-V E2" where V J

Vx.

Now A (A

n

V)U(An I),A

n

6Z and each

xX

A

Vx

62" byheredity. Thus,since {Vxlx6X}isr-locally finite,so is{A

Vxlx

6X}

:_

2, Thus,

xx(A

NVx) A V6 since2"isr-locallyfinite. So A (An V)U(A I)62". Thus,Zisweakly T-local.

THEOREM II.. If

(X,-,2-)

is

Z-paracompact

and 2" is weakly --local, then (X,T

)

is 2"-paracompact.

PROOF. Every opencovercanberefinedbya basicopencover forwhichalocallyfiniterefinement is alocallyfinite refinementoftheoriginalcover. So let/ {U,

Ilc

E A

,U

-, I

6Z}bea basic

--open

coverof X. Then/N {Ulc6

&

is ar-opencoverofXand has ar-locallyfinite

T-openpreciserefinementV {VI6 A which is anZ-cover ofX. Now

V*

{V

Iola

E A is a T-locallyfinite

’-open

preciserefinementof

*

and suchthat is an 2"-coverof X Now {VN

Io1

is ar-locallyfinitesubset of 2" and by weak --locality of 2"; J (Vo Io) 2". Let

X J 6

Z,

thenX J C_ 0

,(oa(V

I))6Z. It remainsonlytoshowthat

V*

isT

-locallyfinite. Butthis is trivial sinceT

c_

-The following corollaryis an immediateconsequenceofTheoremsII.5 andII9.

COROLLARY II.10. If 27 is--local and r-boundary, then(X,-)is2"-paracompact ifand onlyif

(X,T*)

is

Z-paracompact.

T RHAMLETT, D ROSE AND DJANKOVI PROOF. Let

H

{Usla6 A be a7.-opencoverofX. ThenP(L0

open cover ofXand has a 0P(7-))-locally finite (b(r))-open precise refinement W {Wsla6

A

which is an2,-cover ofX. Let V

{Ws

N

Uola

6

A }.

Vis a 7.-open (precise) refinemem of and since (p(7.))_C7., W is 7.-locally finite and so also ]) is 7.-locally finite

By

r-locality,

Ws

(Ws ; Us) _C (Us)

Us

6 sothat

{Ws

(WsAUs)}a6 A is a7.-locallyfinitesubset of 2 and hence the union of this family is a member of 2.

But,

X-t3 (WstqUs) C_

(X

-aE/U

Ws)t3

(s/x(Ws

(WsNUs))) 2"sothat

;

isan2,-coverand(X,T)is2-paracompact. I-!

COROLLARYII.12. If 2"is7.-local and7.-boundary,thenthe following are equivalent.

(1)

(X,(b(7.)))

is2.-paracompact.

(2) (X,7.)is2-paracompact. (3)

(X,7.*)

is2"-paracompact

PROOF. (1) (2)byTheorem II.11 and(2)isequivalentto (3)by CorollaryII.10 To show

(2) (1),letb/=

{ff.ls)la

6

A

beabasic(p(7.))-opencoverof X. Thenb/isa7.-opencoverofX

and hence has a7--open7--locallyfiniteprecise refinement

;

{Vola

A

suchthatX U

;

62"

Let aPO;) {/;(Vo)la A

}.

Each

Vs

C_ Q3s)hence(Vs)C_ /,(Us)) P(Uo) (since2. 7.),thus

0;) is a (p(7.))-open refinement of H. Since

Vs

C_

P(Vs)

for every a, we have

X- U);)C_X- U); _{2; i.e., P0;)} is an2.-cover. Toshow thatb0;) is (7-))-locallyfinite, let

xX.

There exists U7.(x) such that

Uf3Vs

for a

{0,

a2 a,}. We claim that

UNVs=

which implies Unp(Vo) q). _Indeed,

ifUnVs

and UNVs)- 0, then

UNb(Vs)

c_

aP(Vs)

Vs

62(since2" 7.),which comradiets the7--boundary assumption of2.. The following is anexample ofan

Z-paracompact

space (actually paraeompact) (X,7.), suchthat

(X, (p(7.)))is not2"-paracompact.

EXAMPLE. Let X

R

with7-theusualtopology. Let2"

((0,3))

{A

_C

XIA

_

(0,3)} For

_every

U 6 7.,(U) U U(0,3) In particular, for any open set G in

(b(’r)),

(0,3) _CG Let H {(- n,3)ln

N}

t.J{(0,n)ln

e

N},whereNdenotes the naturalnumbers, andobservethatL/isa

(ap(7.))-opencoverofXwiththe property that no finiteopenrefinementorb/cancoverall ofXwith the exception ofsomesubsetof(0,3). Also,noinfiniteopenrefinementofb/canbelocallyfinite sinceevery opensetin<p(7.)>contains(0,3). Thus,(X,<b(7.)>)is not2,-paracompact. Sincethe ideal 2. is7--local but not 7--boundary, we seethat the 7--boundaryassumption cannotbe omitted inCorollary II.12 for

(2)

(1)

Recallthat if(X,7.)isaspace,thenU TiscalledregularopenifU

Int(Cl(U)).

Theregularopen

subsets formabasisfor atopology called thesemiregularization of7., dnoted7-s. We remarkthat if

(X,7.,2")is aspacewith2 7.and

A/’(7.)

C_ 2, then(p(r)) C__7-s[7]. If,inaddition, 2"is’-boundary,then

<()>

COROLLARYILl3. Let (X,7.,2) be a spacewith2 7., 2.7.-boundary, and

A/’(7.)

C_ 2. Then

(X,7.)is2"paracompact iff(X,7-s)is2.-paracompact. I’-i

Aspace(X,7.) is said tobesemiregularif7. _%. Atopological property is called 8emiregularifthe

property isalways shared byatopologyand itssemiregularization. Apropertyiscalled semi-topological if it ispreservedbysemi-homeomorphisminthesense ofCrossley andHildebrand 10] In 11],Harnlett andRoseshow that thesemi-topological propertiesarepreciselythepropertiessharedby7.and

7.*(A/’(7.))

(7-*(A/’(7-))

is denoted by

-

in the literature). Zahid observes in [1], that para-H-closedness is a

semiregularproperty. Since

T

isboth asemiregularandsemi-topological property, a stronger result follows. Asaconsequence, para-H-closednessisalsoasemi-topological property.

IDEAL

(1) (X,-)isalmost paracompact

(2) (X,’)is

A/’(-)-paracompact.

(3) (X,%)is

A/’(-)-paracompact.

(4) (X,’s)is

A/’(-s)-paracompact

(5) (X,’s)isalmost paracompact.

(6) (X,’r)isAf(’)-paracompact. (7) (X,’ra)isalmost paracompact.

PROOF. ForeachAC_X,since%_C%

CLA

C_

CLsA

so that

IntsClA

C_

Int,CL,

A Butforeach

--closed FC_

X, InbF

Inb,

F. Thus,

Inb, ClA

InbCLA

C_

Int,CLsA

for each AC_ X, and

A/’(’s)

C

A/’(-).

Also,

A/’(-)N

’s _C

A/’(-)f

-

{)}

implies that

A/’(-)

and

Af(-s)

are eachboth

--boundary and %-boundary. Now if(X,-) is almost paracompact, (X,-) is A/’(-)-paracompact by TheoremII.1 (1),so thatby Corollary II.12,(X,%)isAf(-)-paracompact. ByTheoremII4,since

A/’(-)

is %-boundary,

(X,%)

is almost paracompact, and therefore by Theorem II.l (1), (X,%) is

A/’(%)-paracompact.

Conversely, if (X,%) is almost paracompact and therefore Af(%)-paracompact, then since

At(%)

c_

A/’(’),

by Theorem II.2,(X,%)isA/’(’)-paracompact. Thenby Corollary II.13,(X,-)is

A/’(-)-paracompact and hence(X,-)isalmost paracompact.

Since

A/’(-)

is--localand--boundaryand since

A/’(’r

)

A/’(-*(A/’(-))

A/’(-),

by Corollary I110,

(’X,-)is almostparacompactiff(X,’r) isalmost paracompact. So almostparacompactness is a semi-topological property Since the

T2

axiomisbothasemiregular and semi-topologicalproperty, so is para-H-closedness I"l

Acollection

A

ofsubsetsofaspace(X,-)is said tobe or-locallyfiniteif

A

LI

An

where each

.Aa

is n=l

alocallyfinitefamily Zahid shows that a

T2

spaceispara-H-closediffevery opencover /2 ofthe space has a c-locally finite refinement

;

t

Vn

such that X t Int(Cl(LI;n)). This result is

n=l n=l

generalizedinthe followingtheorem.

THEOREMII.15. Let (X,-,2.)beaspace with

Af(-)

c_

2", and 2 --boundary. Then(X,%2-)is

2"-paracompactiffeveryopencover L/ofXhas ac-locallyfiniterefinement)2 L

Vn

suchthatX U Int

n=l n=!

Cl(O

PROOF. Necessityis obvious. Toshowsufficiency,let/2beanopencoverofXandsuppose//has aa-locallyfinite refinement

;

t3

’n

suchthatX tAInt Cl(U);n). Let

On

Ul;n so thatX

n=l nffil

OIntCl(On). Let

P1

O1,and

Pn

On

1

Oi)*

forn

>

1. Let

n

{VN

PnIV

;n

foreach n

n--!

1,2,3 and let

n__U|

n

Observe that isanopen refinementof);and henceL/. Weclaimthat is a locallyfinitefamily. Indeed, letx X,andlet

nx

min{n:x Int Cl(On)} Thenx Int

Cl(Onx)

andInt

Cl(Onx)

Pn

for every n

>

nx

i.e.,

Pn

On

(.U

O,)*

and Int Cl(Onx)C_

O*

n [7] Thus (Int

Cl(On))

N LIn) )for everyn

> nx.

Foreachn 1,2 nx,xhas aneighborhood

Gn

-(x)such that

Gn

intersects at most finitely many members of

n

Thus (Int

Cl(Onx))

G

Gn

is a neighborhood ofx which intersectsat mostfinitely manymembersof

.

Weconclude theproof by showing thatX-U

A/’(-)

Weproceedby showing (1) X _n|

P’

and

(2) X _C U

)*.

The result then follows from the fact that

O*

)_C CI(U) t

O

Af(-)

(1) Byassumption,X LI IntCl(On),andIntCl(On)C_

On

since2"isr-boundary Letx EXand let

n--I

mx=min{nlxOn},thenx60*m-

(O

Or)

C*

P’m,

ThusXC_ L’

Pn

438 (2)

TR HAMLETT, D ROSE AND D. JANKOVI(

(uO"

[u(.

n)]"

.__u,

u

.)1"

_

u [u.]"

.--,

vU.

(v

.--u

[P.

n

v

.Vl]

u

[P.

no.]"

rl=l

=UP

n=l

=X t-1

Recallthataspace(X,r)is a BairespaceiffDf(r)N2"

{{3}"

i.e.,Ad(r)isr-boundary.

COROLLARY11.16. Let (X,r,Z)beaspacewith2" r-boundary and

A/’(r)

C_ 2". Then(X,r)is 2"-paracompact iff(X,r)isalmost paracompact Inparticular, if(X,r)is a Bairespace,then

(X,r)

is

.A4(r)-paracompact iff(X,r)isalmost paracompact.

PROOF. TheoremII.15providesacommonequivalentconditionfor(X,r)tobe2.-paracompact El

Insemi-regularspaces,

Z-paracompactness

withrespecttoar-boundaryidealcanbecharacterized as follows.

TIOREM11.17. Let(X,r,2.)be semiregular with2" r-boundary. Then(X,r)is2.-paracompactiff

every regular opencover b ofX has a locally finite refinement

A

(not necessarily open) suchthat

X- uAe2..

PROOF. Necessityisobvious. Toshow sufficiency,let/2 _{U,,Ia

e

A bearegular opencover ofXandassume4 {AlczE A is apreciselocallyfiniterefinementofNsuchthatX- U

.A

E2". For each e

A,

we have

A

C_

Uo

and hence q(Ua) Ua[7, Theorem 5, (5)]. Now l) {(A)lczE A is anopenrefinememorb(andX UVC_X U

A

E2" Toshow 12islocally finite, letx X ThereexistsU r(x)suchthatUfl

A

{3forc czl,c2

en

Observe thatUfl

A

{3 whichimpliesUf-1(A) {3;i.e., ify UandV r(y),thenV

A

_

VflU 2"sothat y (A)

ThusUfl(A) {3 fora {czl,c2,

an

I-1

Thefollowing corollary applies the previous theoremtothe idealofnowhere densesets.

THEOREMH.I$. _{Let (X,r)}bea(Hausdorff) space. Then(X,r) is,almost_paracompact (para-H-closed)iffeveryregular opencoverofXhas alocallyfiniterefinement, not necessarilyopen, whose union isdenseinX.

PROOF. The necessityis clearsince a cover of analmostparacompact (para-H-closed) space by

regular open sets is an open coverand since locallyfinite families are closure preserving For the

sufficiency,by TheoremII14 it isenoughtoshow that(X,rs)is

YV’(r)-paracompact.

But byhypothesis, every regularopencover4 ofXhasar-locallyfinite refinement,4suchthatX U

A

A(r)

Since

rs

C_%&4 islocallyfinite with respect to

rs

and since(X,rs)issemiregular, by Theorem II. 17, (X,rs)is

YV’(r)-paracompact

IH. PRESERVATION

BY

FUNCTIONS AND PRODUCTS

ItwasshownbyMichael in 14]thatthe closedcominuousimage ofaparacompact(Hausdorff)space isparacompact andZahidhas shown thataperfect (cominuous, closed, compactfibers)image ofa para-H-closed spaceispara-H-closedinthe category of Hausdorff spaces. Inthis more

general

setting weofferthefollowing result. First, for anyfunctionf X Yand subsetA_C

X,

let

f#(A)

{y

PARACOMPACTNESS WITH RESPECT TO ANIDEAL

THEOREM HI.I. Let f:(X,T,T)--, (Y,cL,.7")be a continuous openclosed surjection with

fq(y)

compactfor every yEYandf(T)

c_

,.

If(X,T,Z)is

F-paracompact,

then(Y,cz,,7")is

J-paracompact.

PROOF. Let {UIoE A be anopencoverof Y. Then

{f’l(ua)la

A is anopencoverofX

and hence there exists alocally finitepreciserefinement {Vala

A

of

{f’l(Uo)la

A

such that

X- L3

Vo

_{’. Now {f(V)la}E

A

is apreciseopen refinement of{Uo[a

A

andY f(X) f((

aU/

Vs)t I) t.J f(Va)U f(I) so that Y t.J

f(Vs)

Cf(I)

e ft.

To show that

{f(V)[aE

A

islocally finite, let y E

Y;

then there exists anopen setO suchthat

fq(y)

C_Oand

Oq

Vs

0fora {al,c2 c,}. Now

f#

(O)Nf(Vs)

0

impliesOq

Vo

#

_{0 Hence f# (O)is an} open neighborhood of ywhich intersectsatmostfinitelymanysetsfrom the collection {f(Vs)[cE Z

The theorems ofMichaeland Zahid mentionedaboveare sharperin theirspecial case settings than what the previous theorem provides. The previous theoremthough does lead to some meaningful

consequences.

COROLLARYHI.2. Letf.

(X,r,T)

(Y,c,ff)

beahomeomorphism withfiT)C_

ff

If(X,r)is

’-paracompact

then(Y,cx)is,]-paracompact. I-!

Inthelanguageof 11],

’-paracompact

is a"*-topological" property.

Wewillsay thatafunction f: (X,r.Z) (Y,c,ff)is p-continuous iff

f:(X,r,)

-,

(Y,(b(c0))

_is

continuous. Certainlyeverycontinuous function is

@continuous

since

(b(c))

C_cand theconverseis

ot

true. Weremark that the almostcontinuousfunctionsof SingalandSingal [15]are a special case where

,

isthe nowhere denseideal onthespace (Y,cQ.

Weremark that it isclear from the proof ofZahid’sresult (that perfectimages of para-H-closed spacesarepara-H-closedinthe category of Hausdorffspaces)thatit issufficientfor the functiontobe almostcontinuous0p-continuouswith respecttothe idealofnowheredensesetsonthe co-domain).

Itis well known thatperfect preimages ofparacompactspacesare paracompact 16] and Zahid shows thatperfect preimages of para-H-closedspacesarepara-H-closed inthe category ofHausdorff spaces. Weremark that hisproofshows thatperfectpreimagesof almost paracompactspacesarealmost paracompact. In this spiritwe have the followingresult. Given a function f: (X,7-)--, (Y,a,,Y’), we denote by

(fq(,f))

theidealgenerated by preimages of members of if,i.e.

(fq(ff))

{AIA

C_

fq(J)

for someJ

TIIEOREMIII.3. Let f: (X,r,ff) (Y,r,ff)be a perfectfunctionfrom aspace Xontoa paracompactspace Y,with

(fq

(if))

_c

Z. then(X,r)is2"-paracompact.

PROOF. LetL/= _{U,[r A be anopencoverofX. Let."

{F

C_

A

IF

isfinite} andlet

Uv

U

Us

forF ’. Let/A’

{Uv[F

’}

Observethat

{f(Uv)[F

E

.’}

isanopencoverofY

Indeed, ify Y then

f-l(y)

iscompact implies thereexists a finitesubcollection

{Us

,Uo}

such that

fq(y)

C_

t

Uo,.

LettingF {a a,} wehavey

f#(UF)

Now,since(Y,g)isff-paracompact,

I--I

thereexists apreciseopenlocallyfinite refinement

{V[F

’}

of

{f(Uv)[F

’}

such thatY

VF)L)JforsomeJE,.7. Let1,’

{fq(V)

Us[F

"

anda F}. Then1;is anopenrefinementofL/

and we claim: (1));islocally finite, and(2)1;_{is an}Z-coverofX Toshow(1),let x X Thenthere exists V r(f(x)) such that V

V

for finitely many members F of ’. Now observe that

f

(V)q fq_{(Vv) iffV}

V

₀

_{showing that}fq

(V)

intersectsatmostfinitely manymembers of

;.

Toshow

(2),

observe thatforevery FE

’,

fq(V:)

U:

IF

where

I

C_

f-

(J).

NowforF

"

and

a

F,

wehavefq_(V)

Us

_{(Uv Iv)}q

Uo

Us

I.

_{Hence X}

;

X td

{Uo

Iv]F

anda

F}

C_ L{I[FE

’}

C_

fq(J),

l"l

It is well known that the product of two paraompact spaces is not necessarily paracompact

T R HANILETT, D ROSE AND DJANKOVI

spaceis paracompact. Zahidshows in[1 that theproduct ofapara-H-closed space andan H-closed space $]ispara-H-closed. Inthisspirit,weofferthefollowing result.

COROLLARY!11.4. Let

(X,r,2")

be an

Z-paracompact

Hausdorffspace, let(Y,r)bea compact

space, and let

p:X

x Y--,X be the projection function. If ,7 is an ideal on XxY such that

(p-1

(2"))

C_ ,.7,thenXxYisff-paracompact.

PROOF. Theprojection functionp" XxY isperfect. Theresult follows immediatelythen from Theorem III.3 E!

IV. SUBSETS

If 2"is an ideal on anonemptyspace(X,-)andAC_X,wedenote the restriction of 2toAby

2-1A

{I

f3

AII

E

2"}

{B

C_

AIB

E

2"}.

We say thatAis an_2-paracompact

subset

ifforevery opencoverb(of

Athereexists alocallyfinite(with respectto7-)openrefinement

;

ofL/suchthatA UV 2. If27

},then the definition ofAbeinga"{ }-paracompactsubset" coincides with the definition ofAbeing an

"c-paracompact"

subsetin

[19].

Wewill sayAis an_2"-paracompactsubspaceif(X,

rlA,2IA)

is 2--paracompactasasubspace,where

rlA

isthe usual subspace topology. The definitionofAbeinga

"{

}-paraeompactsubspace"coincides withAbeinga"/3-paracompact"subsetin 19].

THEOREMIV.I. IfA

c_

(X,r,2")is an

-paracompact

subset, thenAisan2"-paracompact subspace PROOF. LetL/=

{Uo

f3

Ala

/x bea7-lA-opencoverofAwhere

Uo

7-for eacha

.

Then {Uola E / _{isar-opencoverofAandhencehasar-open"r-locallyfiniteprecise refinement} {V,IcE

A}

suchthat A- U{Vola

A}E2-.

NowV=

{VoNAIc A}

is a rlA-open

rlA-locallyfinite refinementof/,/andA t.J); A t2{Volc A 2-. El

The converse of theabove theoremisfalseasshown byanexample ofan }-paracompactsubspace

(/3-paracompact subset)which isnotan

{

}-paracompactsubset(c-paracompact subset)in

[19].

Zahid defines asubsetAofaHausdorffspace (X,r) tobepara-H-closedif it ispara-H-closedas a

subspaee;i.e., if(A,rlA)ispara-H-closedand henceif(A,rlA) is

A/’(rlA)-paracompact.

Observe that

A/’(rlA)

C_

A/’(r)IA

but the reverse inclusion may not hold. It is shown in [20], however that

.A/’(r)IA

_C

JV’(rIA),

and henceA/’(-)IA

A/’(rIA

ifA_CCI(Int(CI(A))). Thuswehave the following theorem.

THEOREMIV.2. IfA_C(X,r) is apara-H-closed subspace,then(A,rlA)is aA/’(r)-paracompact

subspace. The converse istrueifAC_CI(Im(CI(A))) I"1 Wehavethe following diagram:

.A/’(r)-paracompactsubset Para-H-closed subspace

A/’(r)-paracompactsubspace

Figure

Paracompactness

iswellknown tobe closed hereditary(infact

Fo

subsetsofparacompact Hausdorff spaces areparacompact as subspaces), but Zahid [1] providesan examplewhich showsthat even

H-closed spaces mayhaveclosed subsetswhich arenotpara-H-closedsubspaces

ANIDEAL

PROOF. LetL/= {UslaE A and

Us

E

7}

beanopencoverof A. Then {Usla A j_{(X A)} is a T-open cover of X and hence there exists a T-open precise T-locally finite refinement

{Vsla

A

L

{V}

(Vs C_

Us

andVC_X A)such thatX [VU

(snVs)]

Z. Now A L

Vs

A [VU

(seAVs)]U

C_X

[V

U

(seAVs)],U

henceA

se/\U

Vs

62"bythe heredity of 27. l"i

WeseefromTheorem IV.3thattheexample ofZahid 1]ofaclosed subset ofanH-closed space(and

hence anA/’(-)-paracompactHausdorffspace)which isnotapara-H-closed subspace,is anexampleof an

Af(T)-paracompact subset (and hence an

.M(T)-paracompact

subspace) which is not a para-H-closed subspace

THEOREMIV.4. Let

(X,T,T)

beaHausdorff space. IfAC_Xis an2--paracompact subset,thenA

is

T*-closed.

PROOF. Letx EX-A. Foreach y6

A,

let

Uy

6T(x),

Vy

6T(y) such that

Uy

f

Vy

andnote that x CI(Vy) Now {Vyly A}is aT-opencover ofAand hence there exists apreciseT-open

7-locallyfiniterefmemem

{Vly

EA} of

{Vyly

A}such thatA- JV’

e

2". Nowx

t

CI(V)

for

yEA y

each y impliesx

tyA

CI(V,)

CI(yUAV,

). Let U X-

CI(yUAV)

and letJ A-

CI(yAV,

C_

A-I. ThenU J6

T*(X)

and

(U

J)

N A

,

henceAis

T*-closed.

["l

V)C_

A- U

Vy

yea yEA

The following example exhibits a {0)-paracompact subspace (and hence para-H-closed subspace) which is not anN’(r)-paracompactsubset, thus showing thatnoneof thearrows inFigure are reversible dthat"Af(r)-paracompactsubset"and"para-H-closed subspace"areindependent concepts.

EXAMPLE. Let Xdenote the real numbers andletQ denote therationalnumbers. Letrbe the topology generated by taking the usualopensubsets and {q)lq Q}as asubbase. NowQis discrete and hence paracompactas asubspace, but Qis not

r’(A/’(r))

(="r) closedand hencenotan

A/’(r)-paracompact subset.

Let(X,r)beatopological space. Itiswellknown thatfor everyA_CX,

A’(,d(r))

isregular closed

[5]. More generally,itfollowsfrom Theorems3.2and3.3of[9]thatif2 is acompatibleidealonXwith

A/’(r)

_C 2",then

A’(2)

isregular closed. Thisfact isusedin thefollowing decompositiontheoremfor 2"-paracompact spaces.

TREOREMIV.5. Let (X,r,2-)bean2-paracompactspacewith2 rand

N’(r)

C_ 2" ThenX AU where Ais aregularclosed almost paracompact subspace (i.e. (A,rlA)is

Af(rlA)-paracompact)

and E2". If(X,T)isI-Iausdorff,thenAispara-H-closed.

PROOF. Since2" r,X X 2andfromtheaboveremarks wehavethat

X"

isregularclosed. WeletA

X"

and X

X’.

NotethatX

X*

t

{U

flU

2-},

and since

X"

Cl(Int(X))

we have that

2-IX

is

rlX-boundary.

Nowby Theorem IV.3,

X"

isan

Z-paracomact

subspace,i.e.

X"

isan

2-1X’-paracompact

subspace. Also observe thatsince

X"

isregularclosed,we have

A/’(r)IX"

A/’(rlX"

X

Thus we have

2.IX"

is a

TIX*-boundary

idealon with

A/’(TIX*)

c_

2"IX"

andhence, by Corollary II. 12,

(X,’,rIX)

isalmostparacompactas asubspace. If(X,r)isHausdorff,then

X"

isHausdorffand hence is

para-H-closed.

COROLLARYIV.6. Let (X,r) be an,,(r)-paracompactspace. Then X AU where Ais a regularclosed almost paracompact subspaceand ismeager. If(X,r)isHausdorff thenAis

para-H-closed as asubspace.

PROOF. Itis well known[6,Banach

Category Theorem]

that.Ad(r) r. The result thenfollows immediately fromTheoremIV5

ACKNOWLEDGMENT. The second andthirdauthorsreceivedpartial supportthrough anEast

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