* Topological properties

(1)

Internat. J. &

VOL. 13 NO. 3

(1990)

507-512

*-TOPOLOGICAL

PROPERTIES

T.R.

HAMLETT

and

DAVID

ROSE

Department

of Mathematics

EastCentral University Ada,Oklahoma 74820

(Received June 12, 1989 and in revised form December 4, 1989)

ABSTRACT.

An

zdealonasetX isanonempty collection of subsets ofXclosed under the operatio,s of (heredity)and finite unions (additivity). Givenatopological space

(X,v)

an ideal on X and A

C_X,

/’(A) defined as

U{UEv:U-AE3}.

A

topology, denoted

r*,

finer than isgenerated by the basis

{U-I:U(r, IE3},

,,d atopology, denoted

<(r)>,

coarser than r isgenerated by thebasis

(v)

{(U):UEv}.

(X,r,3)

denotesatopologicalspace

(X,r)

withanideal onX.

A

bijection

f:(X,r,3)

(Y,a,})

iscailc(i

*-homeomorphistn if

f:(X,r*)

(Y,a*)

is a homeomorphism, and is called a -ho,conmri)lis if

f:(X,<

(r)>)

(Y,<(a)>)

isahomeomorphism. Properties preserved by

*-honmomorphisms

arc stu(li,(I

well ms necessary and sufficient conditons for a -honeomorphism to be a

*-homeomorphism.

The sci-homeomorphisms and semi-topological properties ofCrossleyand Hildebrand

[Fund.

Math., LXXIV (197’2),

2:3-254]

areshown to be specialcase.

KEYWORDS AND PHRASES. Ideal, regular open, semi-open, semi-homeomorphism, semi-tol)ological property, semiregular, compatible ideal, topological property,

*-topological

property, r-boundary ideal,,owhre

densesets,meager sets.

1980AMS

SUBJECT

CLASSIFICATION CODES.

Primary:54C10; Secondary: 54D25,54D30.

l.

INTRODUCTION

Given a topological space

(X,r),

a nonempty collection of subsets on

X

is called au zdeal

[l l]

iftim following hold:

I. If

A

and

BC_A,

then

B

(heredity);and

II.

If

A

and

BE,

then

AtJBtl

(finite

additivity).

An

ideal is calledatr-deal if thefollowingholds:

III. If

{An:n=1,2,3

isacountable subcollection of3,then

O{hn:n=l,2,3

}

(countable

a(lditivity). The notation

(X,r,)

denotesanonempty set

X,

atopology ron

X,

andan ideal on X. Given I)oi,t

xX,

wedenoteby

r(x)

the"r neighborhoodsystemat

x’"

i.e.,

r(x)

{Ur:xEU}.

Givenaspace

(X,v,3)

andasubseti of

X,

wedenoteby

h*(3,v)

{xX:Uti3

for every

UCv(x)},

tl,c

local

function

of

A

wthrespectto and

[21].

Whennoambiguityispresent,wesimply write

A*

for

A*(3,’).

Welet

CI*(A)

AtJA*

whichdefinesaKuratowskiclosure operator foratopology

v*(3)

finerthan

(i.e.

vC_’*(3)),

alsodenotedsimplyas

v*

whennoambiguity ispresent.

A

basis

(3,v)

for

v*

can bedescril)c(!

follows

[22]:

(2)

Given spaces

(X,r,l), (Y,a,}),

and afunction

f:(X,r,l)

(Y,a,J),

wc willcall

*-homcomovphsm

respect to

r,t,a,

and } if

f:(X,r*)

(Y,a*)

is ahomeomorphism, or siml)ly

*-homeomorphtsm

wlw,,

ambiguityispresent.

A

topological property

P

will be calleda

*-topologtcal

prol,crly wtlhrespect to

r,,tr,

attl

]

if it ispreserved byany

*-homeomorphism

withrespect tor,l,trand

]

or, followi,got,rconvction, silly

*-topologtcal propertywhennoambiguityispresent.

In this paper westudy

*-topological

properties and show that the senai-topological properties ofCrossh’y and Hildebrand

[4]

are aspecialcase.

Givenaspace

(X,r,)

and

AC_X,

wedenoteby

Int(A)

and

CI(A)

the interior and closure ofA witl to respectively, and by

Int*(A)

and

CI*(A)

theinteriorandclosure ofA withrespectto

r*

rcslwctivcly. Wc abbreviate "if andouly if" by "iff", and use thesymbol

"---"

to mean "implies" or"which itplies"

context. Ofcoursethesymbol --,"_is_{also used}_to_{denote functional}_{correspondence;}_i.e. "f:A---B". 2. *-HOMEOMORPHISMS AND

SEMIREGULAR

PROPERTIES

Since

*-topological

propertiesaredefinedasthosepreserved by

*-honaeonaorphisns,

sufficientconditio,.-for afunction tobea

*-homeomorplfism

areacentralissue.

DEFINITION.

A

space

(X,r,3)

is said to be 3-compact

[14, 19]

if for every open cover

{Uc:aA}

ofX, thereexistsafinite subcollection

{Ui:i=l,2

n}

suchthat

X-U{Uai:i=l,2

n}l.

Observe that whenever isanidealon

X

andf:X-,Yisafunction, then

f(3)

{f(1):li}

isa,idealo,Y. The followingtheorem gives sufficient conditions forafunctiontobea

*-homeomorphism.

THEOREM 2.1.

[6]

Let

f:(X,r,3)-,(Y,tr)

be bijection with

(X,r)

l-compact and

(Y,tr)

llausdorff, if

f:(X,r*)

(Y,tr)

iscontinuous,then isa

*-homeomorphism

withrespectto r,l,tr,and

f(l).

Thefollowingtheorem shows that:l-compactnessis

*-topological

propertywithrespectto and

f(l).

THEOREM2.2. Let

f:(X,r,t)

(Y,r,f(5))

bea

*-homeomorphism.

Then

(X,r)

is

t-compact

iff

(Y,tr)is

f(3)-compact.

PROOF.

NECESSITY.

Assume

(X,T)

is:l-compact and let

{Va:cA}

be aa-open coverofY. Then

{f’l(va):cA}

isa

r*-open

cover ofX. It is shown in

[14]

that

(X,r)

is 3-compact iff

(X,r*)is

Thusthere existsafinite subcollection

{f’l(vcri):i=l,2,...,n

such that

X-Uf-I(voi

Il. Co|lsequcntly,

Y-UVai

f(I)f(l)

andit isshown that

(Y,tr)

is

f(l)-compact.

SUFFICIENCY.

Assume

(Y,r)

is

f(:l)-compact

and let

{Utr:aA}

be a r-open cover of X. Then

{f(Uc):trA}

iaa

tr*-open

coverof

Y,

andthereexists finite subcollection

{f(Uti):i=l,2

n}

such that

Y-Uf(Utri)

f(l)f(l).

Then

X-UUai

Ifil, and theproofiscomplete.

Givenaspace

(X,’),

recall thatasubset

U

of

X

issaid tobe regularopen if

U

Int(Cl(U)).

We

denote by

RO(X,r)

the collection of allregularopen subsets of

(X,r).

The collection

RO(X,r)

is abasisforatopology coarserthan r,denoted rs,called the

semtre9ulartzatton

of

r.

DEFINITION.

Givenafunction

f:(X,r)

(Y,tr)

fromaspace

(X,r)

toaspace

(Y,tr),

is said to bea

6-homeomorphtsm

[15]

iff

f:(X,rs)

(Y,trs)

is ahomeomorphism.

A

topological property

P

will becalled a

semtregular topologtcalpropertl iff it ispreserved by6-homeomorphisms.

Another approachtosemiregular topologicalpropertiesisto definethem to betopologicalproperties shared by topologies which have the samesemiregularizations. This approach is easily seen to be equivalent to the approachin this paper.

Givenaspace

(X,r,),

issaid to be

r-bounda [14]

ifrf’15

{g}.

THEOREM

2.3

[9,

Theorem

6.4].

Let

(X,r,l)

beaspace withrf3

{$},

then

(r*)s.

Using the previoustheorem,wecanshowthefollowing.

THEOREM

2.4. Let

f:(X,r,l)

(Y,tr,])

bea

*-homeomorphism

withrf3l

{t}

and trf3]

{t}.

Then anysemi-regularpropertyis a

*-topological

property.

PROOF.

Let P be a semi-regular property and assume

(X,r)

is P. Then

(X,rs)

is P by definition

--,(X,(r*)s

is

P

by Theorem2.3

(X,r*)

is

P

bydefinition

(Y,*)

is Psincesemi-regular propertiesare

(3)

Wedenoteby(r) tilt;ideal of,owlcredensesetswith resl)ccttor. Give. spaces(X,r)atd *-topological properties with resin’or to r.(r),

,

and

(a)

will be called ot-lopolo!ltcal propcrtt,

r*(Jq(r))

iscommonly known asthe-topologyin theliterature,andisdenotedr

[1fi,17].

Anotherapproachtoa-topological propertiis todefinethemtobetopologicalprolwrticssharedI)

.

Thisapproachiseasilyseen tobe equivalent totheonetakenin thispaper.

Asaneasycorollarytothe previoustheorem,weobtainthefollowingresult of Jankovic’a,d

Rcill.v.

COROLLARY2.5

[10]

Semircgular propertiesareit-topological properties.

PROOF.

Givenaspace(X,T),observe that

(r)Clr

{}

andapplyTheorem2.4.

The list of semircgular properties which have been established in tile literature is qttilc cxlet...ic includes: tlausdorff, Urysohn, almost regular, connected, cxtremally di.connccted, II-closed, S-clo...ed. Igl conpact, andpseudocompact.

3. TIIE

LIFTING

THEOREM

Given a space

(X,r),

a subset

A

of

X

is said to be semi-open

[12]

if there exists a U(ir sucl

UC_AC_CI(U)

or, equivalently, if

AC_Ci(Int(A)). A

function

f:(X,r)

(Y,a)issaid to be 1,re-sem,-Ol,C,,

[I]

if for every semi-open set

A C_X, f(A)

issemi-open in

Y;

and issaidto be trresolute

[4]

if for every se,li-()t),’set

BC_Y,

fl(B)

issemi-open in X. A bijection

f:(X,r)

(Y,a)

is said to be sem,-bomeo,,,O,’l,h,s,,,

[.1]

if,t i.-_

both I)rc-semi-open and irresolute. Propertiespreserved bysemi-homeolnorphismsarcsaid tobe semt-lol,oloqral propevltes

[4].

Itcan be shown thatsemi-topological propertiesarea-topological propertiesas aconsequence ofTheorem,

2.6 of

[4]

andTheorem 2of

[3].

We will establish thisfact (specifically wewillshow that tilesemi-topological propertiesareprecisely thec-topological properties)as acorollarytothe Lifting Theoren proveni, thissectio,.

First.howeverweneed several preliminaryresults.

In

[13]

Natkaniec defines an operator

qa(%r):q(X)

r, where

(X,r,:t)

is aspace and

(X)de,ores

t.l,e

power set of

X,

as follows: for every

AC_X, tb(l,r)(A)

{x:

there exists a

Ur(x)

such that

U-A$}

a,d

observes that

!b($,r)(A)

X-(X-A)*.

We

denote

b(l,r)

simply by

b

when no ambigt,ity is prese.t. ’l’he operator

b

hasbeen studied in

[7]

wherethefollowingisobserved:

(A)

U{Ur:U-AI}.

Notethat

(A)

isopenforevery

A_X.

THEOREM

3.1. Givenaspace

(X,r,:I),

r*($)

{ACX:AC:(A)}.

PROOF.

Denote

{AC_X:AC(A)}

bya. First, weshowthata isatopology. Observe that

)C_(g)

and

XC:_O(X)

X.

Now

if A,B6a,

ACIBC_:_(A)FI(B)

(ACIB)

AtqBqtr. If

{Atr:a6A}C_cr,

tt,e,,

AoC_O(UAa)

foreverya---.

UAaC_O(UAa),

andwehave shown thataisatopology.

Nowif

Ur*,

and

xU,

thereexists a

Vqr(x)

and

I1

such that

xV-IC_:_U.

Clearly

V-UC_:_I

sothat V-Ulbyheredity, and hence

x(U).

Thus

UC__(U)

andwehave shown

r*C-_a.

Now let

Aa.

We

have bydefinitionthat

AC_(A)

AC_X-(X-A)* (X-A)*C:_X-A

X-A

is

r*-closed

and hence

A

fir*.

Thusa

r*

and theproofiscomplete.

It is interesting to observe the the specific form of

A*(Jq(r),r)

CI(Int(CI(A)))

for

AC_X

[2],

and, consequently,in thiscase wehave

(A)

Int(Cl(Int(A))).

Itisknownthat

A*((r),r)

isregularclosed where

.At(r)

denotes theideal ofmeager sets, and hence

(A)

isregularopenin thiscase.

(4)

Itisknownthat

JC(r)...r,

and

.(r).--r

(this

isknownastheBanach

T|IEOREM3.2.

[7]

Let

(X,r,l)

beaspace. Then l..-r iff

(A)-AEI

for every

AC_X.

The followingresultis astraightforwardconsequence oftheprevious theorem.

THEOREM

3.3.

Let (X,r,l)be

aspace withl.r. Then

(A)

U{(U):UEr, /,(U)-AI}.

PROOF.

Denote

LI{/,(U):Ur, (U)-AI}

by

el(A).

Clearly,

I(A)C_/,(A).

Now let x’,(A) i,nplies there exists

UEr(x)

such that U-A(EI.

By

Theorem 3.1,

UC_/,(U),

and

,(U)-AC_[,(U)-U]t[U-A]

(U)-AEI.

Hence

xt(A)

and the proofiscomplete.

Observe that if f:X --.Yis aninjection and

]

is anidealon

Y,

then

f-l(])

{f-l(.1):.l}}

isa,idealo,X. If

(X,r,l)

isaspace and isabasisforatopologyon

X,

then

()

isabasis foratopologyo.

X

c(mrser

thanr

[7].

Denote by

<()>

thetopology generated by

()=

{(B):BE}.

Note

that theprevioustheorem shows that

(J,r)

(J,<(J,r)(r)>),

if l-r.

TIIEOREM

3.4. Let

(X,r,l)

and

(Y,a,})

be spaces with

f:(X,r)

(Y,<(a)>)

a conti,uousinjcctio., }..-a,and

f’l(})C_l.

Then

(f(A))C_f((A))

for everyACX.

PROOF. Let

y(E(f(A))

where

AC_X.

Then by Theorem3.3, thereexists

Va

s,ch thaty/,(V) and

(v)-f(A)e}.

Now

we have

r((v))er(rl(y))

with

f’I[(V)-f(A)]EI

f’I((V))-AI

rl(y)e(A)

yf((A)),

and the proofiscomplete.

TIIEOREM

3.5. Let

(X,r,l)

and

(Y,a,})

be spaces with

f:(X,<b(r)>)

(Y,a,})

anol)en bijectio,, l--.r,

and

f(l)C_}.

Then

f(b(A))C_g,(f(A))

forevery

AC_X.

PROOF. Let AC_X

andlet

yf(b(A)).

Then

f(y)Ef(A)

thereexists Vqr such that

r(y)q)(V)

(V)-A

by Theorem 3.3. Now

f(C(V))a(y)

and

f((V))-f(A)

fI(V)-A]Ef()C_}.

Thus

yW(f(A)),

theproofiscomplete.

THEOREM

3.6.

Let

f:(X,r,J)

(Y,a,})

beabijection with

f()

}.

Then thefollowingareequivale,t:

(1)

isa

*-homeomorphism;

(2)

f(A*)

[f(A)]*

forevery

AC_X;

and

(3)

f(b(A))

(f(A))

for every

AC_X.

PROOF.

(1)

(2).

Let ACX. Assume

yf(A*).

This implies

f’l(y)A*,

and hence there exists

Ur(t-l(y))

such that Uf’IA(5I. Consequently

f(U)a*(y)

nd

f(U)f’lf(A)E}

yf(A)*(],a*)

f(A)*(},a).

Thus

[f(A)]*

C_f(A*).

Nowassume

y[f(A)]*.

This implies there existsa

Va(y)

suchthat

VCIf(A)]

f’l(v)r*(f’l(y))

and

t-I(v)f’IAI

f’l(y)A*(l,r*)

A*(l,r)

yf(A*).

Ilence

f(A*)C_[f(A)]*

and

(2)

holds.

(2)

(3).

Let

AC_X.

Then

f(b(A))

f[X-(X-A)*]

Y-f(X-A)*

Y-(Y-f(A))*

k(f(A)).

(3)

(1).

Let

UEr*.

Then

UC_(U)

by Theorem 3.1

f(U)C-_f(b(U))

k(f(U))

f(U)a*,

hence

f:(X,r*)

(Y,a*)

isopen.Similarly,

f’l:(y,a*)

(X,r*)

is openand isa

*-homeomorphism.

DEFINITION.

A

function

f:(X,r,l)

(Y,a,})

willbecalledab-homeomorph,smw,threspect

o

r, and

} (simply

a -homeomorphsm when no ambiguity is

present)

iff

f:(X,<k(r)>)

(Y,<k(a)>)

is a

homeomorphism.

THEOREM3.7. Let

(X,r,J)

beaspace, then

<,#(r*)>

<(r)>.

PROOF.

Note that for every

Ur

andforevery I1, wehave

k(U-I)

b(U).

Consequently,

b(/3)

b(r)

and

<b(fl)>

<b(r)>.

It

followsdirectlyfrom Theorem 11 of

[7]

that

<b(fl)>

<(r*)>,

hencethe theoremisproved.

Our

nexttheoremisthe maintheorem ofthis section.

THEOREM

3.8.

(Lifting

Theorem).

Let

f:(X,r,l)

(Y,a,})

beabijection with

f(l)

}.

(1)

If isa

*-homeomorphism,

then isab-homeomorphism.

(5)

511

PROOF.

(1)

Assutne

f:(X,r*)

(Y,a*)

is homeomorphism, and let /(U) I)e basic Ol)(,l! .,,et ill <P(’)>with

UEr.

Then f(P(U)) (f(U))by Theorem3.6

f(/,(U))E,(tr*),

I>,t

<g,(rr*)>

</,(a)>

I,y

Theorem 3.7. Thus

f:(X,</,(r)>)

(Y,<g,(tr)>)isopen. Similarily,

f’l:(Y,<,(cr)>}

(X,<,(r)>)is and isa b-hon]eomorpltisln.

(2)

A.sume

is a,-lmnmomorphism, t.ien

f((A))

t/,(f(A))

for every

AC_X

by’[’heoretns3.4 al 3.5. Thus isa

*-homeonmrphism

I>y

Tleorem 3.6, andthe proofiscomplete.

Thehypothesesof1-,-and

-a

arenecessary in

(2)

ofthe abovetheoremasthe following cxatnple shows.

EXAMPLE.

Let

X

be timnaturaln,mbers.

Denote

by n the initialset.ofnaturalnumbers to."i.e.,

{1,2

n}.

Let denote thetopology

{,X}O{In:nX }.

Let

lf

denote the ideal offi,itesul)set.s of X, al observe that

(A)

X

for every

A C_X

so that

<b(’)>

is indiscrete. Also observe that

r*

i tlw li.,,crcte

topologyon X. Let

Y

denotethe natural numbers and lettr bethe indiscrete_topology o,

Y,

a,d cosiler identity function i"

(X,r,3f)

(Y,t,lf).

Clearly is ab-homeomorphisn since<p(r)>and

<(o’)>

are

indiscrete.

However,

tr*

isthe co-finitetopologyandhence isnota -homeomorphsn.

Note that in the above example we have

Jf

not compatible with r but wedo have

3f,

showing that COml)atibilitycannotberelaxedinthehypothesesofTheorem 3.8,

(2),

oneitherthedomainorrange. Also thatthecompatilility hypothesis

-a

cannotbe relaxed to the weaker conditon of/=

tr*

in the ra,ge.

Thenextexamplesl]owsthat tle hypothesis

f(l)

inTheorem3.8,

(1),

cannotbe relaxedto

f(l)C_,.

EXAMPLE.

Let X

{0, 1},

r

{0,

X,

{0}},

a

{0, X},

1=

{O},

J

{O, {1}}

an,!

f:(X,r,:l)

I)e the identity function. Clearly,

f(l)C_J,

l--.v,and

J.--a.

It isalso easilyseen that

r*(l)

a*(J)

aml, hence, is

*-homeonmrphism. However, <(,r)(r)>

v:/:a

<(J,)(a)>

so ti,at is not

’-honeomorphism.

As

an _{application of the Lifting}

Theorem,

we will prove a theorem partially due to Crossley and Hildebrand

([3]

and

[4]),

asmentioned earlier. Firstweneedapreliminary result.

THEOREM

3.9.

Let

f:

(X,’)

(Y,tr)

beasemihomeomorphism,then isa6-homeomorphism.

PROOF.

Let

f:(X,’)

(Y,tQ

beasemihomeomorphism. Itsuffices toshowthat preservesregularopen .sets. If

VC_X

isregularopen, V is semiclopen

(i.e.

bothsemiopen and semiclosed in thesense that

X-V

isalso

semiopen)

so that

f(V)

is semiclopen. Then

lnt(f(V))

is regular open and

CI(f(V))

Ci(Int(f(V)))

a,d so

Cl(f(V))-Int(f(V))

isnowhere dense. Sincesemihomeomorphismspreservenowhere densesets

[4],

B-A

isnowhere densewith

B

trl(cl(f(V)))

and

A

f’l(Int(f(V))).

Thus,

Int(B)-Cl(A)

g

showi,g that

Int(B)C_Int(Cl(A)).

But

Int(f(V))is

semiclopen

A

is semiclopen and

fat(A)

Int(CI(A)).

Therefore,

Int(B)CInt(Cl(A))

Int(A)C_VC_Int(B)

and V

Int(A)C_A.

This yields

f(V)C_f(A)

Int(f(V))

sothat

f(V)

isopen. Since

f(V)

is

alsosemiclosed,itmustberegularopen.

THEOREM

3.10. Let

f:(X,v)

(Y,tr)

beabijection. Thenfisasemihomeomorphism iff is an t-homeomorphism.

PROOF.

NECESSITY. Let

.N’(-)

and

.N’(o’)

denotethe ideals of nowhere dense sets with respect to

"

and

t,respectively.

It

iswell known

[21]

that

.N’(-)"

and

X()--..

Alsoobserve that

<,p(’)>

’s and

<()>

as

[7].

Thusiffisasemi-homeomorphism, itfollowsfromTheorem 3.9that is a-homeomorphism, andit follows from the Lifting Theorem that isanct-homeomorphism.

SUFFICIENCY.

Let

f:(X,’)

(Y,tr)

be an cr-homeomorphism. Then

fa:(X,’a)

(y,.tr)

is a

homeomorphismwhere

fa(x)

f(x)

for each

xX.

Since

fa

and

(fa)-I

preservesemiopen sets and the semiopen subsets of

(X,

-a)

and

(Y,a)

are precisely those of

(X,’)

and

(Y,)

respectively, and

f-1

also preserve semiopen sets.

4.

a-TOPOLOGICAL PROPERTIES

(6)

sharethesamenowheredenseandmeagersetsand that Bairenessis an a-topologicalprol)erty. Since

(X,r)

and (X,

"c)

also share the same dense sets, resolvability and separability are also o-tol)ological I)rOl)erties. lit tim literature, many authors haveisolatedsemitopological properties which in factweresemireg,lar prol)erties ald hence by Corollary 2.5are c-topological and hencesemitopological. Theexamples below show that Baircncss, resolvability, andseparability"aresemitopologicalproperties whicharenotseniregular.

EXAMPLE.

Thespace

X

{1,2,3

with the cofinitetopology isnot Bairesince finite sets are,owhere

dense and

X

ismeager.

Yet

thesemiregularizationofXis Baire sinceit isindiscrete.

EXAMPLE.

Two-point Sierpinskispace is not resolvable whereasitssemiregularization is i,di,r,t,altd thusresolvable.

EXAMPLE.

Theuncountablesetof real numbers

R

withthecocountable topologyis notscparabh,, Il,t.its indiscretesemiregularizationisseparable.

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1.

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D.

Some

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3.

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4.

CROSSLEY, S.G. AND

HILDEBRAND,

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T.

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On

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AND

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AND

REILLY,

I. L.

On

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11.

KURATOWSKI, K.

Topologie

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1933.

12.

LEVINE,

N.

L.

Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70

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13.

NATKANIEC, T.

On-continuityand-semicontinuity points,Math. Slovaca,

36(1986),

No.

3, 297-312. 14.

NEWCOMB, R. L.

Topologies with arecompact moduloanideal, Ph.D. dissertation, Univ. ofCal. at

Santa Barbara, 1967.

15.

NOIRI, T. A

note on

-continuous

functions,

J.

Korean Math.

Soc_.,

18, No. 1,

(1981),

37-42. 16.

NJASTAD, O. On

someclasses of neaxlyopensets,Pacific

J.

Math., 15

(1965),

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17. Remarksontopologiesdefinedby localproperties,Avh, NorskeVid. Akad,Oslo

I(N.S.),

8(1966),

1-16.

18.

OXTOBY,

J.C. Measureand

Category,

Springer-Verlag,1980.

19.

RANCIN, D.

V. Compactness moduloanideal,SovietMath. Dokl.,

13(1972),

No. 1.

20.

SEMADENI,

Z. Functions with sets of points of discontinuitybelongingtoafixed ideal,Fund. Math., L[I

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