On θ regular spaces

VOL. 17 NO. 4 (1994) 687-692

ON O-REGULAR SPACES

MARTIN M.KOV,R

z

Del)artment(ffMathematics Facultyof Electrical Engineering

Technical University fBrn

Are1

VUT,

KravlH)ra

21/XV

602 00Brn(

Czech Republic

(Received June 16, 1993)

ABSTRACT. Inthis paperwestudy 0-regularity anditsrelationsto other topological properties.

We showthat the concepts of 0-regularity

(Jankovi,

1985)

andpoint paracompactness(Boyte,

1973)

coincide. Regular, stronglylocally compactorparaconpactspacesare0-regulax. Wediscuss the i)roblemwhen a

(c()untat)ly)

0-regular space is regular, strongly locally coral)act compact,

orparacompact. We also studysone basic properties ofsubspacesof a 0-regularspace. Some

applications:

A

space is paracomt)act iffthe space iscountably 0-regular and semiparacompact.

A

generalized F-subspaceofaparacompact spaceis1)araconpact iff thesubspaceis countably 0-regular.

KEY

WORDS AND PHRASES. 0-regularity, point paracompactness, covers,filterbases, nets, 0-closure, 0-clusterpoint.

1992 AMS SUBJECT CLASSIFICATION CODES. Primary 54A20, 54D20; Secondary 54D10, 54B05,54D45, 54D30.

1. PRELIMINARIES

AND INTRODUCTION.

In

a topological space

X

a point x

X

is in the0-closure of a set

A

C_

X

(x

cla A)

if every closed neighborhood

(nbd)

ofx intersects A. Thepoint x belongs to the 0-interiorof

A

(x

int

A)

if x has a closed nbd G C_ A. Theset

A

is called 0-closed

(0-open)

if

A

cl

A

(A

inte

A).

A

filterbase (I) in

X

hasa0-cluster pointx X if x

f"l

{cl F]

F

(I)}.

Thefilter base

0-convergestoits0-limitx ifforeveryclosednbd

H

ofx thereisF (I) suchthat

F

C_H.

A

net

(B,

>)

hasa0-cluster point

(a

0-limit)

x

X

if x isa0-cluster point

(a

0-limit)

of thederived filterbase

{{()1

c

>/}

I/

B}.

For any set

S,

wedenoteby

ISI

the cardinality ofS. Thecharacter

x(X)

ofa topological

space

X

wedefineas the least infinitecardinalm such that every point x

X

has anbd base v. satisfying

I’1

<

m. For afamily(I) C_2

X,

wedenote by (I)F the family ofallfiniteunionsof members of(I).

cmt.ainingmlymefthe

lints

.:,y. Thel,ints,r,y are

T-Selaralh,

ifthey haveopenlisjint hinds.

In

this nt.ation, thesptce

X

is saidt,]e

S [3]

if ew.ry twoT0-separablepoints ofXare

T-s,’paral,le. We say that the space

X

is

R

[4]

iffor ,:very x,y

X

satisfying cl

{x}

cl

{y}

thesetscl

{x},

cl

{y}

have disj,,int nbds. Of,servethat any

T

Sl,a,’eis $2 and eachSl,aCeis

S

if andonlyifitis

R.

A

topological space

X

is calledstrongly locally cmapact

[4]

if every x

X

has a chased cmpact nld. Ifevery pint of a topological space

X

has a nbd base consisting of open sets with c,,mpact ],,undary, then

X

is said t,, ]e rimcompact

[3].

A

space is almost c,,ml,act

[3]

(,,r

abs,,lutely H-closed

[1]

,,r

H(i)

[4]

)if

_{every open}filter base in

X

hms a cluster l,,,int, or

equivalently, if every filter be in

X

has a 0-cluster point, or

Mso

equivalently, if every open

’verof

X

has afinite subfamily whoseunion is dense inX.

T

see that these conditims are

really equivalent, werefer thereader to

[3]; [4]

and

[8].

Finally,a topologicMspace is said to be semiparacompact

[7]

if everyopencoverof thespacehasaa-locallyfinite open refinement.

The aim of this paper is to study two independently introduced concepts, 8-regularity

(.Jankovi6

[4]

)and

point

(countable)I,

aracoml,actness(Boyte

[1] ).

Thefirstconcept,defined in

termsoffilter bases,wasusedby D. Jankovidtoextend the closedgraphtheorem ofD. A.

Rose.

Wesay that a topologicalspace

X

is

(countably)

8-regularifevery

(countable)

filter bme in

X

witha8-cluster point hasacluster point.

The second concept, point

(countable)

laracompactness, w defined by J. M.

Boyte

to improveseveral covering theorems

(e.g.

the well-knownMichel’s

Theorem,

characterizing

pro’a-compactness by open a-locallyfinite refinements ofopen

covers)

in absence ofmy sepm’ation axiomsuch

T2

orregulm’ity.

A

topologicalspace

X

issMd tobe point

(countably)

pm’acomp-act if forevery open

(countable)

coverfl of

X

andeachx

X

thereisanopen refinementflof such that is locMlyfinite at x.

In

this paperweshow that

(countable)

8-regulm’ity and point

(countable)

paracompactness

coincide. We generalize some

Boyte’s

d Jkovid’s results concerning the problem when a

(countably)

8-regularspace is regular, strongly locMly conpactorpacompact. We

Mso

derive

anecessyd sufficient conditionfor agenerMized Fa-subspaceofapacompact space to be

paconpactifnoadditionM sepationiom is sumed.

Weinust note that J. Chewin

[2]

also studied

severM

relationsofpacompactness d full normality.

Among

thegn, the "localparcompactness

(1)"

actuMlycoincideswithpoint

pa-compactnessof]. M.

Boyte.

2. MAIN RESULTS.

Our starting pointisthefollowing theorem:

THEOREM 1. Let

X

be atopologicMspace, m w acdinal. Thefollowing statements

eequilent:

(i)

For every opencover fl of

X,

[fl[

m, deach x

e X

there is closednbd Gofx such that Gcanbe coveredbyafinitesubfily of ft.

(ii)

Foreveryopencoverfl of

X, [[

m, and each x

X

thereis open refinement

’

of such thatfl islocallyfiniteatx.

(iii)

Forevery filterbe in

X, [[

m, havingnocluster point and for echz

X

thereare

F

dopendisjoint sets

U,

V

such that x Uand

F

V.

(iv) Every

filterbe in

X,

[]

m,witha8-clusterpointha cluster point.

PROOF. Suppose

(i).

Letflbe opencoverof

X,

[fl[

m, andletx X. Thereisaclosed nbdGofxdaitesubfaly

F

such that

G

g

ver

U. Let

Itis clearthat Q’ isanpen refinement ofQ andGmeets at mostfinitely manyelements ofl’.

Itfollows

(ii).

Sul)l)ose (ii).

Let l)eafilterbasein

X

with11(,chtster point,

[1

-<

_m,and let.r X. The collection Q

{X

\elF

F

,I}

is anopen directedc,,ver of

X

with

Ifl

_<

-.

By (ii),

there existsanpen refinementf’ which is locallyfiniteat x. Thenx hasa nbd U intersecting only

afinitenumber of members ofQ’. Let V

U

{s]

s

fi

Q’,s

n

u

}.

Wehave

U

n

V and sincefisdirected, thereissome

F

cI,such that

X

\VC_

U

s]s

Ft’,

S U

#

c_

X

\elF.

Itfillowsthat el

F

C_

V,

and hence

(iii)

is proven.

(iii)

implies

(iv)

trivially.

Suppose

(iv).

Let Q _beanopen coverof

X,

]f[

_<

m, and let x X. IfX

QF,

we are finished. Let

X

fF.

It follows that the family ,I,

{X

\

UI

U 9tF is a closedfilter base havingnocluster point and satisfying

]q,]

<

m.

By (iv),

the point x cannotbea0-cluster p,,int ofq,. Itfollows that thereis aclosed nbd Gofx and U

fF

such that G

(X

\

U)

,

i.e.

GC_

U.

Thus

(i)

is fulfilled. That completes theproofof thetheorem.

REMARK

1. Consider thefollowingconditionforaspace

X

andacardinalm

_>

w"

(v)

Every 0-convergentnet

o(a,

>)

in

X,

having

]a]

_<

rn, hasacluster point.

Toseethat

(v)

isweaker than

(iv)

ofTheorem 1,wecantake thefilterbasenaturallyderived from

0(A,

>

).

Analogically, if

x(X)

_<

rn,wecanshow that

(v)

implies

(iv).

Let ,I bea filterbase in

X,

]I’ _<

m, havinga0-clusterpointx X. Let

r

beanbdbase ofzwith

Ivl

_<

m. Thecollection

F

{F n

cl

U]

F

I,,

U

r

}

is a filter base with

]F] _<

m. Using Axiom of Choice, we can construct axnap0

F

[.Jer

Gsatisfying

o(G)

GforeveryG F. Then

0(F, C_)

is anet

0-convergingtox

and,

consequently, havingacluster pointby

(v).

Itfollows that hasacluster point,whichimplies that

(iv)

isfulfilled.

COROLLARY 1.

A

topologicalspace is point

(countably)

paracompact if andonly if the space is

(countably)

0-regular.

Obviously,everyregularspace is0-regular. D.Jankovidprovedin

[4]

thatall rimconpactor strongly locallycompactspacesare 0-regular. That resultcanbe easilyseenfroxn the condition

(i)

of Theorem1. Fromthecoincidencebetween 0-regularity andpoint paracolnpactness itfollows that the class of0-regularspacesalsocontainsparaconpactspaces. Conversely,we canask when

a0-regularspace isregular, strongly locally compactorparacompact.

DEFINITION1. Letm,nbecardinals.

A

topologicalspace is saidtobe

(rn, n)-cover

regular

ifforevery open covertof

X, ]fll

-<

m, and each point x fi

X

thereisaclosed nbd ofx which canbecovered byasubfamily 9t C_ f such that

]f]

<

n.

In

temnsofthisdefinition,thespaces, satisfyinganyoneof the conditionsof Theorem 1,are

(m,

w)-cover

regular.

THEOREM2. LetXbeatopologicalspace,m acardinalnmnbersuch thatw

_<

x(X)

_<

m.

Then

X

is regularif andonlyif

X

is

S

and

(m,w)-cover

regular.

PROOF. The necessity is clear. Toseethe sufficiency, let U C_

X

be anopen set and let x U. Let

v

beanbd base ofxsuch that

]v _<

m. Sincewesupposethat

X

is$2, thecollection fl

{U}

tO

{X

\cl

Y]

Y

r,}

is anopencoverof

X,

If] _<

m. Hencethereis a closed nbdGof

(U’

(x

,,:iv;))

D,.oi. g

n

(19’

and

V, V:,...

V

,’r, such that

C::

U

U

=

_i=

COROLLARY4

(Boyte).

In

T2

spaces, pointparacompactnessis equivalenttoregularity.

THEOREM3.

A

topologicalspace

X

isstronglylocallycompact ifandonlyif

X

islocally

compact and0-regular.

PROOF. Thenecessity is clear. Conversely, if

X

islocally compact, thereexists a family I, ofcompact sets such that

{intK[KeO}

is an open cover ofX. Let

X

be

0-regu-lar. Then every x

X

has a closed nbd G and cotnpact sets K,K2,...K, O such that

GC__

[.J:’=

intK, C_

[.J:’__

K,. SinceGis closed,itfollows that Giscompact. Thus

X

is strongly locally compact.

The condition causing paracompactness ofa

(countably)

0-regular spacewill bediscusscd later. Becauseevery closed subspaceofa0-regular space isobviously 0-regular,one canexpect that F-subspacesalsohave that property. Thefollowingexampleshows thatit isnotthecase.

EXAMPLE

1. Thereisacotnpact topologicalspace

X

containinganFa-subspace

Y

which is notcountably 0-regular.

PROOF. Let Y

{2,3,...},

U

{n.xln=l,2,...}

for every x

e

Y. The family

{U:]z

Y}

definesa topology

(as

its

base)

onY. Since

V

Vv

for every x,y

Y,

every opennon-emptyset UC_Yhas clyU Y. Itfollowsthat thenet

id(P, >_),

where

P

isthe set of all prime numberswith theirnatural order

_>,

isclearly 0-convergent,but withnocluster point inY. Itfollows from Theorem and Remark1,that

Y

isnotcountably 0-regular.

LetX 1

)

tOYandtakeon

X

thetopologyofAlexaaadroff’s compactification ofY. Tosee

that

Y

isanF,-subspaceof

X,

let

K

Y

\

[.Jv>

Uv

forevery x Y.

Every

K

isclosed, finite, and,hence,compactix,topologyofY. Itfollows

Kr

isclosedinX. Sincex

Kr,

Y

[.joo__2

K.

Thatcotnpletesthe proof.

Let

X

bea topologicalspace. Wedenoteby

o(X)

the family of all sets

B

C

X

such that

clo {z}

C_

B

forevery x B.

For

a 0-regularspaceX, thefollowing theoremshowsthat

a(X)

constitutes anixnportant class ofsubspacesofX.

THEOREM4. Let X beatopologicalspace, rn

>_

w acardinal. Thefollowing statements

arefulfilled:

(i) flo(X)is

acompleteBoolean setalgebra

(ii)

If

X

is

(m,w)-cover

regularand

Y

e o(X)

asubspace,

x(Y)

_<

m, then

Y

isalso

(m,w)-coverregular.

PROOF. To show that

(i)

is true, we must prove that the unions and complements of arbitrary elements of

o(X)

are members of

fo(X).

Obviously, only theproof concerning the complementsis non-trivial.

Let

A

e fBo(X)

andsupposethat

cl {x}

Y

\

A

for some x

Y

\A. Then thereexists

y

A

such that y

cl0

{x}.

It

follows that x, y arenot T-separableand hence x

cl0 {y}.

That is a contradiction, because by the definition of

0(X)

we have

cl

{y}

C

A.

It follows

cl

{z} _c

Y

\

A

forevery x

Y

\

A,

which implies

Y

\

A

fla(X).

Wehave shown

(i).

Now,

let

X

be

(m,w)-cover

regular,

Y

o(X)

and

x(Y)

_<

m. Let

0(A,

>)

be anet in

Y,

[A[ _<

m,which0-convergestoy

Y

inthetopologyof

Y.

Then

0(A,

>)

0-convergesto y in

X

andhence, according to Remark 1,

o(A,/>)

hasa clusterpoint x X. Onecan easily verify thatx,y cannot beT2-separable,whichimplies thatx fi

clo {y}

andhencex Y. Becausex isa

cluster point of

p(A,

>/)

also withrespecttothe induced

topology

of

Y

and

x(Y) _<

m, Remark1

completes the proofof

(ii).

COROLLARY6. Let X be a 0-regulartopoh)gicalspace. TheneachsubsI)ace, whichcan beexpressed byBo()lean _{ol)erations (i.e.} \,

(.J,

)(,f

0-Ol)en

(or

0-closed)

sets, is 0-regulm"as

well.

As an illustrati()n, we present here a theorem which was first proved by

Boyte

[1]

using covering terminology and methods. That result was also independentely derived by Jankovi

[4]

in terms offilter 1)ases and convergence.

An

easy proof of the theorem is an inlmediate consequence ofthedefinitionsof0-regularityaa(lahnost compactness. Weleave it tothe reader.

THEOREM5.

A

topologicalspace iscompact if andonlyifthespace is0-regularand almost compact.

The following theorem describessomerelations between

(countable)

0-regularityand

1)ara-compactness. It slightlyimprovesanalogicalresults of Michael

[6];

Mack

[5]

and

Boyte

[1].

THEOREM6.

A

topologicalspace

X

isparacompactifandonlyif

X

iscountal)ly0-regular

and semiparaconpact.

PROOF. The necessityis clear. Conversely,assumethat

X

issemiparacompact aaxd count-ably 0-regular. Let fl be an opencoverofX. Semiparacompactness ofX implies that fl has

an opena-locallyfiniterefinelnent, say f

U,%l

""

whereevery f, is a locallyfinitefaanily

refiningf. Let U,,

J

{U

U f,,i

_< n}

forevery n N. The family

{U,}neN

isacountable openincreasingcoverof

X

andsince

X

is countably O-regular,thereexists anopencover of

X

whose closures refine

{U,},e

_N. Because

X

is semiparacompact, has anopena-locally finite

refinement,say

O’

,__

0,,consisting oflocallyfinitefamilies 0,. Foreveryn N let

V,=U{BIBe,,clBC_Uj,i+

j

<n}.

(2.1)

Observe that

{V,,

},,eN

is anopenincreasingcoverofX. Becausethe family

(.J,=

q) is locally finite,wehave cl

V,,

_C

Un-.

Finally, for everyn N andU

fin

let

W,.,(U)

U

\cl

V,,.

(2.2)

It can be easily seen that the family

F

{W,,(U)[n

N,U

fl,,}

is an open locally finite refinement of ft. Indeed, for every x

X

let k N be the least index such that x (

U

for

some

U

(

12.

Since cl

V _c

Ua_,

it follows thatx

W(U).

Hence

F

isanopencover, which,

obviously,refines ft. Toseethat

F

islocally finite,let x

X

and let m ( Nbemy indexsuch

that x

Vm.

Because

{V,,}neN

is anincreasing family, we have

V,,,

3

Wn(U)

for every

n

>

m,

U

6

f..

But the family

[.Ji=

f/, islocally finite. Let S be a nbd of x, intersecting at most finitely manyelements of

I.Ji=

f/i. Since forevery 1, 2,...m,

U

f, we have

W,(U)

C_

U,

the set

S

Vm

isanbd of x, meeting only finitelymany setsofthecoverF. Hence

F

islocallyfiniteand therefore

X

iparacompact.

COROLLARY7

(Boyte). A

topologicalspace isparacompactifand onlyifthespace is point

paracompactand semiparacompact.

COROLLARY

8

(Mack).

A

topological space is paracompact if and only if the space is countablyparacompactand semiparacompct.

COROLLARY9

(Boyte). In

aLindelSfspace,point countable paracompactnessisequivalent toparacompactness.

In

atopological space

X,

a subspace

Y

C__

X

is called a generalized F-subspace of

X

if forevery open

U

C_

X

suchthat

Y

C

U

thereis an

F-subset

F

ofX satisfying

Y

C._

F

C_ U.

Singal and Jain

[7]

proved that in a semiparacompact space, every generalizedF-subspace is semiparacolnpact.

However,

Example shows that nothingsixnilarholds forparacompactnessiu

general. But thefollowingresultweobtainfromTheorem6 andthe resultmentionedabove.

THEOREM 7.

A

geueralized Fa-subspaceofa paracompact space is paracompact if and onlyifthe subspaceis countably0-regular.

COROLLARY11

(Mack).

A

generalized Fa-subspaceofaparacompactspace is paraconp-act ifandonlyifthe subspaceis countably paracompact.

Replacing’closed’ by ’0-closed’, weobtain adefinition of aOF-subspace.

In

other words, YC_Xissaidto bea0Fa-subspace ofatopologicalspace

X

if

Y

is acountableunionof 0-closed sets. Since every 0-closed set is closed, we caaa use Theorem 7 and Corollary 6 to obtain the

followingresult.

COROLLARY

12.

A

0Fa-subspaceofaparacolnpact spaceisparacompact.

Finally,inregularspaces,weobtainthewell-knownresult ofMichael

[6]

thatanF-subspace

ofaregularparacompactspace isparacompact.

ACKNOWLEDGEMENT.

The authorisgratefulto ProfessorM.Mrevi(forsendingthecopy of

[4]

and for herhelpfuladviceandcomments. Theauthoralso thanks toMissKathleen Graham for her

language

advice.

Thiswork dedicatetoOur Ladyand xny Mother.

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1.

BOYTE, J.M.

Point

(countable)

paracompactness,

J.

Austral. Math. Soc. 15

(1973),

138-144. 2.

CHEW,

J. Localparacompactness,Portugal. Math. 28

(1969),

103-109.

3.

CS/itSZ./tR,

A.

GeneralTopolo_Ky, AkademiaiKiad6, Budapest, 1978.

4.

JANKOVI,

D. S. /9-regularspaces, preprint, published:

Internat. J.

Math. Sci. 8

(1985),

no.3,615-619.

5.

MACK,

J. Directedcoversandparacompactspaces,Canad. J. Math. 19

(1967),

649-654. 6.

MICHAEL,

E.

A

noteonparacompactspaces,Proc. Am.Math. Soc.4

(1953),

831-838. 7.

SINGAL,

M.

K.;

JAIN,

P. Semiparacompact spaces, Period. Math.

Hungar.

5

(1974),

243-248.