Some counterexamples and properties on generalizations of Lindelöf spaces

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 4 (1996) 737-746

737

SOME

COUNTEREXAMPLES AND

P.R.

OPERTIES

ON

GENERALIZATIONS OF LINDELOF SPACES

FILIPPOCAMMAROTO Dipartimentodi Matematica Universita’ di Catania VialeA Doria,6 95100Catania,ITALY

GRAZIASANTORO DipartimentodiMatematica Universita’ di Messina SalitaSperone,n 31

98166Sant’

Agata,

Messina,ITALY

(ReceivedJune 19, 1992and in revisedform September 27, 1995)

ABSTRACT. Inthispaperwegivesomesignificativecounterexamplestoprovethat some wellknown generalizations ofLindelofpropertyareproper Alsowegivesome newresults,we introduceand study

the almostregular-Lindelof spaces as ageneralization ofthe almost-Lindel0fspaces and as a new and

significative application of the quasi-regular open subsets of ].

KEY WORDS AND PHRASES: Lindelofspace, almost Lindelof,weaklyLindelofandnearlyLindelof, semiregular andalmost-regular space, regularcover

1980AMSSUBJECT CLASSIFICATION CODES: 54C10, 54C20

1. INTRODUCTION

Inliteraturethereareseveral generalizations ofthe notionofLindelofspace[2]andthese are studied

separately for different reasons and purposes In 1959 Frolik [3] introduced the notion of

weakly-Lindel0fspaces that,aPterward,was studiedbyseveralauthors: Comfort,Hindmanand Negrepointis[4]

in 1969, Hager [5] in 1969, Ulmer[6] in 1972, Woods [7] in 1976, Bell, Ginsburg and Woods [8] in 1978 Aboutthistopicin 1982 Balasubramanian[9]introducedandstudiedthe notionof nearly-Lindelof spacesthat is between Lindel0f andweakly-Lindelofspaces In1984 Willardand Dissanayake 10]gave

the notion of almost k-Lindel0fspace, that fork

N0

wecall almost-Lindelof,andthatisbetween the

nearly-Lindelof and weakly-Lindelof spaces. Tobecomplete, it isusefultorecallsomerecentpapers of Pareek 11 which are analmostsurveyof allmaingeneralizations of Lindelofspaces

In this paper we fix our attention on the main generalizations ofLindel0fspaces, e weakly-Lindelof, almost-Lindel0fand nearly-LindelOf spaces Ourpurposeis tostudy therelations between them and some newproperties but, mainly, to constructsomesignificativecounterexamplestoprovethatthe

studiedgeneralizationsareproper.

Moreover, the counterexample 3.1l, provingthat there exist weakly-Lindelofspaces not

almost-Lindelof, guides us to introduce and study a new generalization ofLindelofspaces, e the almost

regular-Lindel0fspaces These almostregular-Lindel0fspacesare a newandsignificative applicationof

quasi-regularopen subsetintroducedbythe first author andLoFaro in 1981

We concludethe paper proposing the study oftwo newand natural generalizations ofthe almost regular-LindelOfspaces, e theweakly regular-Lindelofand thenearly regular-Lindelof spaces

Inparticular,thispaperiscomposed of four parts In we studythenearly-Lindelofspacesas a

generalization of Lindelofspaces(whileBalasubramanianhas studied them as ageneralization ofnearly compact spaces),wegivesomepropertiesand acounterexample ofanearly-Lindel0fnotLindelofspace

738 F CAMMAROI’OANDG SANTORO

weakly-Lindelof spaces and acounterexampleofweakly-Lindelofnotnearly-Lindelof space, moreover,

we study the almost-Lindelofspacesthat arebetween nearly-Lindelofand weakly-Lindelofspaces, we

give the necessary counterexamplesandproperties Inthe last section we introducethenotionsof almost regular-Lindelof,weaklyregular-Lindelof andnearly regular-Lindelofspaces

Wehave thatthefollowing implications hold

Nearly-L Almost-L Weakly-L

NearlyR-L

=

AlmostR-L

=

Weakly R-L

PRELIMINARIES

Throughout the present paperXandYalways denote topological spaces,:ranelement ofXand

theneighborhoodsfilterof:rinX The interiorand the closure of any subset

A

of

X

will bedenoted by

int

(A)

or andcl

(A)

or

-respectively.

If

A

C_ S

c_

X,

then

ints(A)

and

cls(A)

willbe usedtodenote respectively the interior andclosure

ofAinthesubspace S With

{a: }>,

and

{a

}er

wedenote thesetof all elements

a

for each

>_

and for each c Nrespectively

Recall some definitions

DEFINITION 1. Asubset

A

c_

Xiscalledregularly open (resp. regularly closed)if

A

A

(resp

A=A)

Thetopologygenerated by theregularlyopensubsetsof the space

(X,

7-)

isdenoted by7-"andit is

called semiregularization of

X,

if7- 7-*then

X

is said tobe semiregular _12].

DEFINITION2 13]. Atopologicalspace

X

is said tobe almostregulariffor eachz

c X

and each regularly open neighborhood

U

/g, there exists a neighborhood

V

such that

V

C

V

C

Us,

or, equivalently,iffor anyregularlyclosedsetCand any singleton

{z}

disjoint from

C,

there existdisjoint opensetsUandVsuch thatC

c_

Uandz V.

Itis true that aspaceXisregularifandonlyif it issemiregularandalmostregular 13

DEFINITION3 [14]. Atopological space

X

is saidtobe nearly compactifevery opencover

{

Ux

}

x^ of

X

admitsa finitesubfamilysuch that

X

t

DEFINITION4[2] Let

X

beatopologicalspace. AcoverV

{

V}j

of

X

isa

refinement

of anothercoverb/=

{Ux}x^

ifforeachj Jthereexists anA(j)

c A

such that

V

c

DEFINITION5 [2]. A family

{Ux

}XA

of subsets ofatopological space

X

islocallyfimteiffor every point z Xthere exists a neighborhood

U

b/ such that the set

{A

A

U

N

Ux

}

is finite

1.

NEARLY LINDELOF-SPACES

DEFINITION1.1

[9].

Atopologicalspace

X

issaidtobenearly-LmdelOfifforevery opencover

{Ux)xA

ofXthereexists acountable subset

{A,),r

of

A

such that

X

t_J

Ux

_(ie ifeverycoverof

nN

Xby regularly opensetsadmits acountablesubcover).

Itisclear thateverycompact spaceisnearly-Lindel0f,but the converse isnot true(forexamplethe

reallineRisnearly-LindelOfbutit is notnearly compact).

Moreover, every Lindel0fspace is nearly-Lindel0fbut the converse is not true as the following example shows.

EXAMPLE

1.2. Letf be the smallest uncountableordinalnumber and

A

[0,

f).

Theset

A

has the property that for eacha

A

theset

[0, a)

iscountable, while

A

is not. LetX

{a,

q,

a)

where

c A

and j N. Wedefine inXatopologysuchthatthe points

{a)

areisolatedand thefundamental system of neighborhoods ofthepoints

{ c)

and

{a)

are

GENEIM_.IZATIONS()FI.INI)I,:I.( )I:SPACES 739

respectively Xso topologized isHausdorff but notLindelof, infacttheopencover

{B}

tO

{B,

,

},

_A hasnotcountable subcover Ontheotherhand, Xisnearly-Lindelof Infact, let

Ua

a,x bea cover of

X

andAsuchthata E

U

Then

X\

isacountableset Itfollows thatXisnearly-Lindelof I’-!

PROPERTY1.3. A space

(X, 7-)

isnearly-Lindelofifandonlyif

(X, 7-’)

is Lindelof 1-1 COROLLARY1.4. Anearly-Lindelof space

(X,

7-)is Lindelof ifand onlyif it issemiregular l-1 This is animprovement of [prop 5, g]that holdsonly forregularspaces

PROPOSITION l.fi [9] Atopological space

X

isnearly-Lindelofif and only if forany family

{Ca}ac,

by regularlyclosedsetsofX withcountable intersection property, the intersection A

Ca

is

non-empty l-I

PROPOSITION 1.6. Let

X

be an almost regular and nearly-Lindelof space Then for every disjoint regularly closed

C

and

C2

there exist two open sets U and V such that Ufq_V

₀

and

CICU,

C2cV

PROOF. Since Xisalmost regular,for each z E

C

thereexists an open neighborhood

Uz

such that UC2-) We can suppose

Ux

to be regularly open The family

{Ux}xc

2{X\C}

is a

regularly open cover ofX and, since

X

is nearly-Lindelof, there exists a countable set of points

:rl,x2,...,z,,... ofXsuch thatX

(

UrU

)

td

(X\C,

It follows that for each

nNC, c

tJ

U

and

U.

C2

Analogouslythereexists afamilyof regular opensets

{

Vu.

such that

(

)

()

andU= U

Gn,

V= U

H

U

andV D

C

O

Let

G

U,

k

H

Vy,,

k

nN hen

andVsoconstructedarethe opensetsthat we arelooking for

DEFINITION 1.7 15] A space X is said to be nearlyparacompact if

eve

cover ofX by

regularlyopensetsadmits alocallyfinite refinement

PROPOSITION 1.8 Let

X

be an almost regular and nearly-Lindelof space Then

X

is nely

paraeompact

PROOF Let

{Ux}

aA beacoverofXbyregularlyopensets Foreachz

X

and

A

A

such that z

U

there exists anopen neighborhood

U

ofzsuch that

U

c

Ux

Weesupposethat

U

is

relly openso

{U

}x

isa

relar

opencoverof X. Since X isnearly-LindelOf,thereexists a

countablesetof points z, z,..., z,.., ofXsuchthat X U

U.

Foreachn Nchoosea

A

A

nN

such that

U

c

U

and put

g

U

k= By construction

{g}e

is a refinement of

{U}e

dit is alocallyfinitefamily. In fact, letz

X.

Thenthereexist

U,

(since

{U}e

is a coverof

X)

and

U,

such thatz

U,

C

U,.

Wewillprove that

U,

intersectsatmostfiNtelymany

members of thefaly

{

g }e

Since

then

Up

is not contained in

V

for eachr

_>

p

+

Iwhile

Uxp

c

V,.

So

Ux,

fq

V

O

for eachr

>

p

+

1, therefore

U,

intersectsatmost a finitenumberofsets

V,.

I-I

PROPOSITION 1.9. Let

X

be a nearly-Lindel0f spaceand Y a nearly compact space Then

X Yisnearly-Lindelof.

PROOF. Let

{

Ux}aA

be a coverof

X

Ybyregularlyopensets Withoutloss of generality,we cansuppose

Ux

Va

Wa

where

Va

and

Wx

areregularlyopensets in

X

and Y respectively Fix x5X, foreach y

Y

there exists A_u

A

such that

(x,

y)E

Vx

x

Wa

The family

{Wa

}y6y

is a cover of

Y

by regularly opensets and, since Yis nearly compact, it admits a finite subcover Let Y

Wax

t3 t3

Wa

Put

H

Vax

f

Vx

and

r(x)

Au,,...,

Au

}

H

is aregularly openset

of X andhencethe family

{

H

}zex

is aregularlyopencoverofX Since

X

isnearly-Lindelof, there

7/10 F CAMMAROTO AND(; SANTORO

XxY=

(H,,,)

xY= U

(Hr,,xW,):

U

(V,

xW,).

Since the lastmemberis acountable family, wehave that

X

x Yisnearly-Lindelof I-!

REMARK 1.10 In general theproduct oftwonearly-Lindelof spacesis not nearly-Lindelof In

fact, let K be the Sorgenfrey line K is normal, and hence regular, and Lindelof and therefore it is

nearly-Lindelof TheproductKx Kisregular,but it isnotLindel0f[2, 3 815] andthereforeitcannot benearly-Lindelof(seeCorollary 4)

2.

NEARLY-LINDEL(F SUBSPACES ANDSUBSETS

AsubsetS’ofaspace

X

is said tobe nearly-LindelofifSisnearly-LindelofassubspaceofX (ie S

isnearly-Lindel0fwithrespecttotheinductedtopologyfrom thetopologyofX)

DEFINITION2.1 AsubsetSofaspaceXis said tobenearly-LmdelOfrelaOveto

X

iffor every

cover

U

A--A by opensetsofX suchthat S

c

_AAU

Ux,

there exists a countable subset A,,

},,

r

c

A

such thatS

c

U

nf_N

PROPOSITION2.2[9] Let Xbe aspace and

A

anopen subsetofX ThenAisnearly-Lindelof

ifandonlyif it isnearly-Lindelofrelativeto

X

LEMMA 2.3

[9]

Let

B

be aregularly closed subset ofanearly-LindelOf space

X

Then Cis

nearly-Lindel0frelativeto

X

r-!

COROLLARY2.4[9] Aclopen ofanearly-Lindelofspaceisnearly-Lindelof I-!

PROPERTY2.5 Let Xbeanextremallydisconnectedspace(ie the closure ofanopenset isopen

[2])

and,5’C_X IfSisnearly-LindelOf then,5’ isnearly-Lindel0frelativetoX.

PROOF. Let

Ua }Xh

be anopen familyofXsuch thatS

c

U

Ux

Consider

Vx

S

Ux

for

each

A

A,

then

{V}xeA

is an open cover ofS.

By

hypothesis thereexists a countable subfamily

(Vx}er

such thatS U

intscls(Vx

).

Sincefor eachn

IIVA

C

UA,

then

V

C Since hEN

Xisextremallydisconnectedthen

intscls(VA)

C

intxclx(Uo)

clx(U,)

Thisprovesthat C U

hEN

[,,

i.e.Sisnearly-LindelOfrelativetoX. El

REMARK2.6. Ingeneralaclosed subset ofanearly-LindelOf spaceis neithernearly-LindelOfnor

nearly-Lindel0frelative tothe spaceasthe subset

{

c,

},A

inExample1.2shows

PROPOSITION2.7. Let

X

beaspaceand

c

X. The followingareequivalent

(i) Sisnearly-Lindel0frelative to

X;

(ii)for every familyby regularlyopensetsof

X

thatcover

S,

thereexists acountable subfamily coveting

,5’;

() freery

farni

{C}eA b

reguary

csedsets

X

schhat

( AC)

fqS thee

exsts

a countable subset ofindices

{A,},er

C

A

such that

(,rCx,)

fq_{S 0}

PROOF. (i)

=

(ii) Itis obviousbythe definition.

Let

{C,x},XeA

be a regularly closed familyin X suchthat

(fC,x]

N

S=.

Then

()

C

X\(,XAC’x)

,XA

(X\Cx);

hence

{X\C’>,}XA

is aregularly open family coveting

S S, then there

exists acuntable subfamily

{X\Cx"}neN

suchthatSC U

(X\Cx")’

(iii)

:=

(i) Let

{Ux}x

_A be a family by open subsets of X such that S C U

Ux

Then

ScB2

UC

,

therefore

X\

S=0, ie

X\

NS=0

By

AA AA AA

such that f’l

X\

fqS 0 and

hypothesis there exists a countable subfamily

X\

therefore

X

n

S 0,i.e S

c

U

ffx.

Thiscompletestheproof l-I

Gt.NEIGM.IZATIONSOFI.INI)EI.()I:SPACES 7 41 PROPOSITION2.8. Aspace

(X,

"r) isopen hereditarily nearly-LindelofifandonlyifanyAE

-"

is nearly-Lindelof

PROOF. Let B c XbeanopensubsetofX ByProposition 2 2 it is sufficienttoprovethatBis

nearly-LindelofrelativetoX Let

Ua

a,,,x beafamilyby regularly opensetsof

X

such thatB

a(A Theset

A

U

Ua

belongsto-’,soby hypothesisAisnearly-Lindelof Hencethere existsacountable

A-. A

subfamily

Ua, },cr

of

Ua

_{ax such that}

A

1.3

Ua

and thereforeB

c

U

Ua,

Conversely, letXbe hEN

open hereditarily nearly-Lindelof Sincer" C7-,it is obviousthatanyA

THEOREM2.9. Let

f"

X Ybe aclosedcontinuous_{function and, for each V} Y, let

f (V)

benearly-LindelofrelativetoX IfYisnearly-LindelofthenXisnearly-Lindelof

PROOF. Let

{C’a}aA

be afamily ofregularly closed subsets ofX with countable intersection

property Let M

A

TM,

e each#EMisoftheform#

(A,A2,

...,A Put

The family

C’,

_eMis afamilybyclosed subsets of

X

withcountable intersection property and also the family

{f(C)}cM

in Yis so SinceYisnearly-Lindelof, byProposition 5 thereexists EYsuch that

ff

E

f(C)

for each # EM It follows that

f-l()NC’,-7/:

0

for each # E

M,

hence

f-l(ff)

intersectsallcountable intersectionsof

Ca

withAE

A

Since

f-1

(if)

isnearly-LindelofrelativetoX,by Proposition2 7 (iii), we have

(

C’a/

f-1

\aA /

()

:/: 0 and thus

aAf’l

C’a

0 This, by Proposition 5, implies thatXisnearly-Lindelof

REMARK2.10. Recallthat,foratopological space

X,

the

Lmdelofnumber

l(X)

isdefined asthe smallest cardinal number

A

such that every open coverof

X

admits asubcover of cardinality A Itis

naturaltogeneralizethis notiontonearly-Lindelofspace definingthenearly-Lmdelofnumber

of

X

nl(X)

to bethe smallest cardinal number_#such that everyregularlyopencoverofX admits asubcover of cardinality#

Obviously

nl(X) <_ l(X)

and thisinequalitycanbe proper Forthispurpose wecan seeExample

12

3.

ALMOST-LINDELOF

AND

WEAKLY-LINDELOF

SPACES

DEFINITION 3.1 [10] A topological space X is called

almost-LtndelOf

ifevery open cover

Ua

}

aeA ofXadmits acountablesubfamily such thatX U

Ua

DEFINITION3.2[3] Atopological spaceXis saidtobe weakly-Lmdelofifforevery opencover

Ua

}

aeh ofXthere exists a countablesubfamilysuch thatX U

Ua

nN

PROPOSITION 3.3. Atopological spaceX is weakly-Lindelofifandonly iffor any family of closed subsets of

X{

C’x

}

xhsuch that_AA

CA

thereexists acountable subfamily

{

Ca}r

such that

int(,NC’,,)

=0.

PROOF. Let

{C’a}aA

be a family of closed subsets of X such that fq

C’a

₀

Then

X U

(X\C’a)

sobyhypothesis thereexists acountable subfamilysuchthatX U

(X\C’a)

Hence

AEA hEN

X\,r(X\Cx.)=O,

ie

int(X\(N(X\Cx.)))= int(NC,)

0

Conversely, let

{UA}aeA

be

an opencoverofX Then

(X\UA)

0

and therefore thereexistsacountable subfamily such that

AA

int

(,r(X\Ua))=

0.

So

X=

X\int(,I(X\Ua,))

X\(,r(X\Ua))

pROPOSITION3.4. LetXbe atopologicalspace Forthefollowingconditions

742 F CAMMARO’I() ANI)( SANT()RO

(ii) anycover

Ua

a,

.

ofXbyregularly opensetsofXadmits acountable subfamilywithdense union inX,

(iii) if

{C’a

a, A is a family ofregularly closed subsets ofX such that CI

CA

0, then there exists a acA

countable subfamilysuchthatint

(

f3

CA

O,

we have that(i)

=

(ii)

=

(iii)

=

(ii)and ifXis semiregularthen (ii)

=

PROOF. (i)

=

(ii) is obviousfrom the definition Theproofof(ii)

=

(iii) isquite similartothe

proofof Proposition 33 replacingopen coverwith a regularly open cover ofX Wewill provethe

implication (ii)

=

(i)when

X

issemiregular Let

Ua)a

1 beanopencoverofX Byhypothesiswe

can supposeany

Ua

to beregularlyopen Then thereexists acountable subfamily

Ua

},,,

such that to

Ua,,

X Thiscompletestheproof I-1

Obviously, if aspaceisnearly-Lindelof thenit isalmost-Lindelofand ifaspaceisalmost-Lindelof then it isweakly-Lindelof But the following exampleshowsthat weakly-Lindelof propertyor almost-Lindelofpropertydoesnotimplythenearly-Lindelof property

EXAMPLI3.5. Let f be thesmallest uncountableordinal number and

A

[0,

f)

asinExample

2 LetX

{a3,

b

3,c, a,

b}

where E

A

and3EN Consider in

X

thetopologysuchthat the points

a

and

b3

areisolated and the fundamentalsystemofneighborhoodsof thepoints c,

},

a and

b}

are

B

{c,,au,

b,}3>,,

B

{a,a,j},>_o,jer

and

B

{b,b,j},_>o,el

respectively X sotopologizcdisHausdorff and scmircgular butit isnotncarly-Lindclofas wecan scc

considering

{Ua

}a^

beanopencoverof

X

Then there exist

A,

Az

A

suchthata

Ua

andb

Ua

Theset

X\ (Ua

tO

Ua

iscountable,so itfollows easily that

X

isweakly-Lindelof. Notethatthisspace

X

isalso almost-Lindelof.

Belowwe willgivethe constructionofanexample ofaweakly-Lindelofspace thatit is not almost-Lindel0f

PROPOSITION3.6. Atopological spaceXis almost-LindelOf ifand onlyifevery family

CA

of closedsubsetsofXsuchthat fq

CA

O

admits acountable subfamily such that

rCxo

O

AA

PROOF. If

{Ca}a

is a family by closed subsets of

X

such that f3

CA

,

then the family

{X\C,x}xa

is an open cover of X. By hypothesis there exists a countable subfamily such that

tO

X\Ca,

X,

e.,Q_Ta,

.

Conversely,let

(Ua}aa

beanopencoverof

X

Then

{X\Ua}a^

is afamilybyclosedsetssuch that

(X\U,)

By

hypothesis thereexists acountable subfamily such

that q

int(X\Ua,)

,

e X t3

Uao

Thiscompletestheproof.

hEN hEN

PROPOSITION3.7. Let Xbeatopologicalspace Forthe followingconditions

(i)

X

isalmost-LindelOf,

(ii)everyregularlyopencover

{

Ua

}

aAadmitsacountable subfamily such that

X

tO

(iii) every family

{C’a

}aeA

of regularly closed subsets of

X

such that

CA

0

admits a countable subfamilysuchthat CC’_a 0;

wehavethat(i)

=

(ii)

=

(iii)

=

(ii)and if

X

issemiregularthen(ii)

=

(i)

PROOF. (i)

=

(ii)is obviousby thedefinition. Theproofof(ii)

=

(iii)isquitesimilartotheproof

ofProposition3 6replacingopencover with aregularlyopencoverofX Wewillprove the implication

(ii)

=

(i)when

X

issemiregular. Let

{Ua

a beanopencoverofX.

By

hypothesiswe cansuppose thatany

Ua

isregularly open,then there exists a countablesubfamily

Ua,

},r

such that tO

Ua,,

X

(;ENI’RAI,IZA’IIONS()1’I.INI)F,I.()FSPACI!S 743

THEOREM 3.8. A weakly-Lindelof, semiregular and nearly paracompact space X is almost-LindeloF

PROOF. Let

U }

,,x beacoveroFXby regularly opensets SinceXisnearlyparacompact, this cover admits a locally finite refinement

{V}<

I" Since

X

is weakly-Lindelof then there exists a countable subfamily such that X

=V.,

Since the family

{V..,},,:

is locally finite, then

V

V

[2, 11] Choosing, for each

n,

A,

A

suchthat V,

c

U. weobtain

X

,-"

,,N

U,,

By

Proposition37Xis almost-Lindelof

PROPOSITION 3.9 [9] An almost regular space is an almost-Lindelofspace ifand only if it is

nearly-Lindelof

CONSTRUCTION OF A

WEAKLY-LINDELOF

SPACE

LEMMA 3.10. The real line canbe paitioned intheunion ofafamily, of cardinality 2

,

by

countabledenseandpaiise disjoint subsetsof

PROOF. Let

Q

be thesetof therationalnumbers Considerthe following equivalencerelation on z ifand onlyif z-Q.

The pamition of so obtained is the onethatwe want, in fact

eve

equivalence class isof the form

z

+

Q,

wherez

R,

andit is acountable dense subset of

EXAMPLE

3.11. Let bethereallineandrthe usualtopoloonit Bythe previouslemma we can representRas aunionof continuummany countable dense and paiise disjoint subsets of

R

We

canwritethispaitionasN=(S)USo,

wherethesetlhascardinality2 Let

r

be thetopology

onRhaving thebase

{

S }et

R Letabe thetopologygeneratedbyrandT and letX

(R,

a) We

willshowthat

X

isnotalmost-Lindelof Since

S0

iscountable,wecan write

S0

{Zl,

z,..., z,,

...}

Considerthe open cover ofX

x-Suppos

tSat

X

isalmost-indd,tSenthereexists acountableset ],i2,

,,

}

Z suc5that

Xn+

=n

x

u

SincetheLebesguemeasureof theset

eU

[z

,

z

+

has cardinNity greater than

N0

But

X

n[z

_V,z

+

]

C(&

U

S0)

and,sincethesecond member is countable, we obtain a contradiction. Wewill shownowthat X iswey-Lindelof Let

{

U

ebe opencover ofXand

So

{0,

x, x,

...}

asabove. Since inthetopologye

eve

point of

S0

has thesamendamental systemofneighborhoodsasinthetopolor, then for eachn N

there

est

an openset

g

inrandanindex

A

such that

z

g

c

Ux

The set g U

g

is

openinrand

S0

C g

c

U

U.

Let

z

S.

For y r-openneighborhood

(because

S0

isdensein

(N, r))

So g

,

hence

N

g andthisshows thatz,

cl(V)

Weobtainthat

X

clo(V)

clo

U)

and thereforeXisweakly-LindelOf

4.

LOST

VLN-LELF

The previous exple suggests some interesting remarks But before it is usel to recall the

follongdefinitions

EFINITION

4.1 opencover

Ua

}

ae ofatopologicalspace

X

is saidtobe

relar

iffor

eve

A

there exists a non-emptyregularly closed subset ofX such that

Ua

(ie

U

is

744 F CAMMAR()’I’() ANI)(i SANT()R()

DEFINITION 4.2 [l] Atopological spaceXis saidtobeweakly compactifevery regularcover admits a finitesubfamilysuch that the union is densemX

LEMMA4.3. Let Xbe thespaceinExample3 LetCbe aregularly closedandAanopenset such thatC’

c

A Thenint,,

(C’)

c int7

clo (A)

PROOF. We denotewith:r0 and:r, the elementsof

So

andS, respectively Weshow beforethat if :r0 Einto(C) then :r0 E int,

cl,,(A)

Sincethe fundamental systemof neighborhoods of:rt) Isthe same whether in thetopologyaor in7-,thenthelemma istrue Nowlet :r, into(C) Thereexistsar-open neighborhood

V

of:r, such thatV,IqS, c C WewillshowthatV, c cl,

(A)

Let:rt V,and let

V

be an arbitrary a-open, and therefore 7--open, neighborhood of :r0 Since :r0 V,AV0, we have

V,N

V0

:/:

0

andthusS,NV,,

V0

:/:

0

Thisshows thata:0

cl,,(V,

S) CC

c cl,,(A)

Let:rjE V,

Supposethat z

clo (A),

e there exists a7--openneighborhood

V

ofa:j such that

V

Sj

Since:rj

V

AV,, then

V

V, -7/: 0 and therefore

V

tO

So

_{:fi 0} Letz0

V

V,, Wehave seen above that:to Cc AC cl,,(A), since

A

isa-open hence thereexists aa-open, and therefore7--open

neighborhood

V0

of :r0 such that

V0

C

A

Since

V0

V

V

:

0 then, by density of Sj,

V0

V,

V

Sj 0 and therefore

A

V

S-]:

that is a contradiction So it is shown that

:r,

clo (A)

andthereforeV,C

clo

(A)

The proofiscomplete

PROPERTY 4.5. The space XinExample 3 11 satisfiesthe following property every regular

cover

U

AofXadmits acountable subfamily

U,}r

such thatX O

Ux,

PROOF. Let

U

A be aregular cover ofX For any

A

A

thereexists a regularly closed

Ca

c

Ux

such thatX O

into(Ca)

Bythepreviouslemma wehaveX t..Jint-clo(Ux) and,sinceX

AA Ae_A

is Lindelof with respect to the topology "r, there exists a countable subcover such that

X--intclo(UA,,)--,,t.J

clo(UA),

l-l

eN

The previous property suggests us to give a new definition that generalizes the weakly-Lindelof property

DEFINITION 4,6, Atopologicalspace

X

iscalled almostregular-Lmdelofifeveryregularcorot

UA)A

AofXadmits acountablesubfamilysuchthat

X

REMARK 4.7. Obviously almost-Lindelofimplies almost regular-Lindel0f, but the converse in

general is not true, in fact the space

X

in Example 3.11 is almost regular-Lindelofbut not almost-Lindelof

THEOREM4,8, Analmost regular-Lindelof and almost regular space

X

isnearly-Lindelof

PROOF, Let

(UA)AeA

bea coverby regularlyopensetsofX Foreach:r

X

thereexists

Ax

E

A

such that x

UA

SinceXisalmost regular,there existtworegularly open subsets

VA

and

W’

such

thatx(

Vx

C

VA

C

W,xW,x

C

U,x,

[13,Th 22] Thefamily

{W,x}:x

is aregularcoverofXand,

sinceXisalmost regular-LindelOf,there exists a countablesetof pointszl,:r, :r, ofXsuchthat

X t_J

WA

So X t_J

UA

and thereforeXisnearly-Lindelof C!

:rN

The previous theorem implies thefollowing

COROLLARY4.9. Let

X

beanalmostregularspace Then

X

isalmost regular-Lindelofifand

onlyif it isnearly-Lindelof

IZI

Wegivenow acharacterizationof almost regular-Lindelof spaces

THEOREM4.10. AtopologicalspaceXis almostregular-Lindelofifand onlyiffor every family

{CA}A

A of closed subsets of

X,

such that foreach A(

A

there exists anopenset

Ax

Ca

with

AA

,

there exists acountable subfamilysuch that

Cx,

0

AA nN

PROOF. Let

{C’a

aea beafamilyofclosedsetsof X suchthat for each A

A

there exists an

et

A

D

C

with V

A=O

It follows that

X-X\(\

-/

=U(X\--)A.

But, since

open

AA AA

C

C

A

c

x

C

A,

then

X\A,x

C

X\.,,x

C

X\C,x,

and therefore X tA

(X\C,x)

The family

GENFRALIZATIONS()FLINI)EI,OFSPACES 745

X\C:

A,A is aregular cover ofX, sinceX is almost regular-Lindelof, thenthere exists acountable

subfamilysuchthat

and therefore N

,

CA 0 Conversely, let

UA

A,Ix be a regular cover ofX Foreach A EA there exists aregularly closed

CA

ofXsuchthat

CA

C

Ca

C

Ua

and_AcA

Ca

X Thefamily

XUx

x,.x of

closedsetsissuchthat, foreach

a

A,

thereexiststhe openset

XUA

D

XU

and such that

AA AA

By hypothesis, there exists a countable set of indices

{A}v

such that

int(XUx)=,

e

(X)

So

X

U andthereforeX U

Ux

Thiscompletestheproof

nN nN

ALMOSTGULAR-LINDELOF SUBSPACES AND SUBSETS

A subsetS ofaspace

X

is saidtobe almost regular-LindelofifS isalmostregul-Lindelofas a

subspaceof

X

DEFINITION4.11. AsubsetSofaspaceXis saidtobealmost

relar-Llndelofrelattve

toXif

for each family

Ua

_Aof opensets of

X

satising thecondition

SC

U,

and

(.) foreachA

A,

there exists anonemptyregularlyclosedset

C

ofXsuch

thatCcUaandSC

thereexists acountablesubset

{A}s

C

A

suchthatS C nN

THEOM 4.12. IfS is an almost regular-LindelOfsubspace ofaspace

X,

thenS is almost

relar-Lindelof

relativetoX

PROOF. Let

{

Ux}xa

be a cover ofSsatising thecondition(.) ForeachA

A,

wehave that

Cx

Sand

Ux

Sareopensets in

S,

and

Cx

SisclosedinS The family

{Ux

S}xa

isanopen

coverofS Wewill show thatit is a

relar

cover of thesubspace S ForeachA

A,

wehave that

cls(x_

S)

c

Cx

S

c

Ux

S,

where

cls(x_

S)_

isregularlyclosed inS

Moreover,

we have

S=

aS

and C

aScintscls

xS

thus S=

intscls

xS

Since S is an aA

almost

relar-LindelOf

subspace of

X,

thereestsacountable subcover suchthatS

cls(Ux,

S)

Since for each n N

cls(Ux,

S) c

Ux,,

weobtainthat S

c

Ux,.

Ts

showsthat S is almost

relar-Lindel0f

relative toX

TREOM4.13. If

eve

relarlyclosed subset ofaspace

X

is

most

regul-Lindelofrelative to

X,

thenXisalmostregular-LindelOf

PROOF. Let

{

Ux

}

_XA bea

relar

coverof X. Foreach

A

A,

thereexists anonemptyregully

closedset

Ca

of

X

such that

Ca

C

Ux

and

X

U

Ca

Fix

bitra A0

6

A

d let

A"

A{A0}

Put

K

XC<,

thenKisregularlyclosedin

X

andKC

Ca.

Therefore

{Ux}xA.

is a coverof

K

A6A*

byopensetsofXsatisng thecondition(,)ofDefinition 4 11 and hencethereexists acountable subset

{A}s

c

A*such thatKC

Ux

Sowe have

X=KUC=KUU=

ugh=

U,.

Thisshows thatXisalmostregular-Lindelof

COROLLARY 4.14. If

eve

proper regularly closed subset ofa space X is almost

regular-Lindelof, then

X

isalmost regular-Lindelo

THEOM4.15. Let

X

be analmost regular-Lindelof space If

A

is aproperclopensubsetofX,

746

F CAMMAR()’I’( ANI)(; SAN’I’()R()

PROOF. Let

U,

x,A be a coverofAbyopensetsofXsatisfying the condition(.)of Definition 4 11 Thefamily

Ua

}x,

tO

(X\A)

isaregularcoverofX SinceXisalmost regular-Lindelof, there

exists acountablesubfamilysuchthat

ThereforewehaveAC t3

Ua.

Thiscompletestheproof I"1

Weconcludethispaper introducing thefollowingtwodefinitions

DEFINITION4.16. Aspace

X

iscalled weaklyregular-Lmdelofifevery regularcover

U,x

4,

,,

of

X

admits acountable subfamilysuchthat

X

tO

DEFINITION4.17. AspaceXiscalled nearly regular-Lmdelofifevery regularcover

Ua

xAof

Xadmitsacountable subfamilysuch thatX

Obviouslywehavethefollowing implications

Nearly-L =, Almost-L =, Weakly-L

Nearly regular-L Almost regular-L =, Weakly regular-L

We leave open the study of these two newgeneralizations ofLindel6f propertyand the relative

implications

ACKNOWLEDGEMENT. Thisresearch wassupported bya grant fromC.N.R (G N S.A.G.A and

M U R S T through"Fondi40%"

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[2] ENGELKING,

R.,

GeneralTopology,

PWN, Warszawa,

1976

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COMFORT, W.W, HINDMAN,

N.and

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,

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of

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Pactfic

J.Math.79(1978),37-45.

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S.P, Onalmostregularspaces, GlasmkMat. Ser.

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MATUR, A,

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