On spaces whose nowhere dense subsets are scattered

Internat. J. Math. & Math. Sci. VOL. 21 NO. 4 (1998) 735-740

735

ON

SPACES WHOSE NOWHERE DENSE SUBSETS

ARE

SCATI’ERED

JULIAN DONTCHEV DAVID ROSE

Department

of Mathematics University of Helsinki

Hallituskatu15 00014Helsinki 0,

FINLAND

Department

of Mathematics SoutheasternCollege 1000LongfellowBoulevard Lakeland,Florida33801-6099,

USA

(Received

November 12,

1996)

ABSTRACT.

The

am

of this paper is to study the class of N-scattered spaces, i.e. the spaces whose nowhere dense subsets are scattered. The concept was recently used in a decomposition of scatteredness atopological space

(X,

7")

isscattered if andonlyif

X

isa-scattered itsa-topology is

scattered)

andN-scattered.

KEY

WORDS

AND

PHRASES: Scattered space, a-scattered, N-scattered, rim-scattered, a-space, topologicalideal.

1991

AMS SUBJECT

CLASSHClCATION

CODES:

Primary: 54G12, 54G05; Secondary: 54G15, 54G99.

1.

INTRODUCTION

Atopologicalspace

(X, 7-)

is scattered if every nonempty subset of

X

hasanisolated point, e.if

X

has no nonempty dense-in-itselfsubspace. Thea-topologyon

X

denotedbyr

[6]

isthe collectionofall subsetsofthe form

U\N,

where

U

isopenandNisnowhere dense or equivalently allsets

A

satisfying

A

C_Int

IntA.

Ifr 7-,then

X

is saidtobe ana-spaceor anodecspace

[5].

Allsubrnaximaland all globallydisconnectedspacesareexamples of c-spaees. We recall thataspace

X

issubmaximal if every densesetis open andgloballydisconnected if everysetwhich canbeplacedbetween anopensetandits closure isopen

[2].

Recentlya-scattered spaces were considered in 1,

7]

spaceswhose c-topologies are

scattered)

and itwasprovedin

[7]

that aspace

X

isscattered if andonlyif

X

isc-scattered and N-scattered. The aim of this paperis to study indetail the class of N-scattered spaces, i.e. the spaces withall nowheredense subsetsbeingscattered.

Recall thatatopological ideal2", i.e. a nonempty collectionofsets of a space

(X,

7-)

closedunder heredity and finiteadditivity, is r-localif2"contains all subsets of

X

which arelocally in2", wherea subset

A

is saidtobelocallyin2"if it has anopencover each member of which intersects

A

in an ideal amount,i.e. eachpointof

A

has a neighborhood whose intersection with

A

is a member of2". This last condition is equivalentto

A

beingdisjointwith

A"

(2-),

where

A"

(2")

{z

6

X" U

Iq

A

_{2" forevery}

U

6

’x}

with 7"xbeing theopen neighborhoodsystem at a point z6

X.

Formore detailsconcerningthe last concepts werefer the readerto

[4,

9].

2. N-SCATTERED

SPACES

Definition1.

A

topologicalspace

(X,

’)

iscalled N-scatteredifevery nowheredensesubset of

X

is scattered.

Clearly every scattered and every -space,i.e.nodecspace,isN-scattered.

In

particular,all submaximal spacesareN-scattered. Thedensity topologyon therealline isanexample ofanN-scattered spacethat notscattered. Thisfollows straight from 10, Theorem

2.7]

and the factthat thedensity topologyis dense-in-itself The space

(,L)

from Example3.14below showsthat evenscattered

spaces

neednotbe spaces. Another class of spaces thatareN-scattered

(but

only alongwiththe

To

separation)is

Cranster’s

736

..

DONTCHEVANDD. ROSE

THEOREM2.1

[7]. A

topologicalspace

(X,

r)

isscattered ifand onlyif

X

is a-scatteredand N-scattered.

THEOREM2.2. If

(X, v)

is a

T

dense-in-itself space, then

X

isN-scattered

N(r)

S(r),

where

N(T)

is the idealof nowheredensesubsetsof

X,

and

S(r)

isthe family ofscatteredsubsets of

X.

PROOF.

It

isknownthat in a

T-space,

S(’)

isthesmallestlocal ideal containingtheideal

I

offinite sets

[8].

Ifthe spaceisalso dense-in-itself,

I

C_

N(T),

alocal ideal,sothat

S(7")

C_

N(r).

It followsthat ifthe spaceisalso N-scattered, i.e.,

N(T)

C_

S(-),

then

N(v)

S(-).

Ofcourse thislast equationimplies that the space isN-scattered.

EXAMPLE

2.3. Let

X

whave the cofinitetopology

-.

Then

X

is a

T

dense-in-itselfspacewith

N(r)-

I.

Clearly,

X

is an N-scattered space, since

N(-)--I

C_

S(7").

By

Theorem 2.2,

N(r)

S(’).

Also,

X

isfarfrom being

(a)-scattered

having no isolated points. Itmay also be observed that thespace ofthisexampleisN-scattered being ana-space.

REMARK

2.4.

A

space

X

iscalled

Coointwise)

homogeneousiffor anypairofpointsz,

X,

there is ahomeomorphism h:X

X

with

h(z)

.

Topologicalgroupsaresuch spaces. Further, sucha spaceis eithercrowdedor discrete.

For

if oneisolated point exists, then all pointsare isolated.

However,

thespace

X

of Example2.3 is acrowded homogeneousN-scatteredspace.

Noticing that scatteredness and a-scatteredness are finitely productive might suggest that N-scatterednessisfinitelyproductive.

But

this isnotthe case.

EXAIVIPLE

2.5. Let

X

bethespace of Example2.3. Thesubspace

X

x

{0}

_C

X

2_{is homeomorphic}

with

X

a non-scattered space.

However,

X

x

{0}

is nowhere dense in

X

2 _{relativeto the product}

topology. Further,

X

2_{isnoteven rim-scattered. Forif}

B

_{isany open base for the product topology every}

nonempty

V

/3 is a unionof products ofcofinitesetsin

X,

andeverynonempWproduct of twocofinite subsets of

X

is a unionof members of

t3

aeVa,

where each

Va

B

and for eacha

A,

Va

U

er(a)[(XF)

x

(XE)],

where

F

and

E

are finite subsets of

X.

Evidently, 0

E

forall

(a,-},)

/k

F(a).

Now

select any a e/X. The boundary of

Vo

is

Bd(V,,)

CI(V,,)\V,

X2\Va,

since

CI(Vo)

_D

[CI(XF)

x

CI(XF_)]

X

2 _for

any7

e

r(a).

Thus

Bd(V,)

_D

X

x

{0},

since

Va

(X

x

{0})

}. Since

X

x

{0}

isdense-in-itselfas asubspace of

X

2,

and thus alsoas asubspace of

Bd(V,,),

itfollows that

Bd(Vo)

is notscattered, so

X

is notrim-scattered.

EXAMPLE

:2.6. The usualspace of reals,

(R, #)

is rim-scatteredbut not N-scattered. Certainly, the usual base ofbounded open intervals has the property that nonempty boundariesofitsmembers are scattered.

However,

thenowheredense

Cantor

set isdense-in-itself Anotherexample ofarim-scattered spacewhich isnotN-scatteredisconstructed in

[1].

REMARK

2.7.

It

appears thatrim-scatteredness ismuch weaker than N-scatteredness.

THEOREM

:2.8. N-scatterednessishereditary.

PROOF.

Suppose

that

(X,-)

is N-scattered and let

A

be any subspace of

X.

Since

N

(T A)

C_

N

(7) A,

every nowhere dense subsetofthesubspace

A

isscattered.

THEOREM

:2.9. Thefollowingareequivalent:

(a)

Thespace

(X, 7-)

isN-scattered.

(b)

Every

nonempty nowheredense subspacecontains anisolatedpoint.

(c)

Every

nowheredense subsetisscattered, i.e.,

N(7)

C_

S(-).

(d)

Every

closed nowhere dense subsetisscattered.

(e)

Every

nonemptyopen subset hasascattered boundary,i.e.

Bd(U)

(f)

The

T-bounda

of every a-opensetisr-scattered. (g) The boundary of everynonempty semi-open set isscattered.

(h)

There is abase forthetopology consisting ofN-scattered open subspaces. (i) The space hasanopen cover ofN-scattered subspaces.

(j)

Every

nonemptyopen subspaceisN-scattered.

(k)

Every

nowhere densesubsetisa-scattered.

ON SPACES WHOSE NOWHERE DENSESUBSETSARESCATTERED 737

let

U\N

7-,

where

U

7" and

N

N(7").

We

may assume that

N

C_

U.

Then

Bd(U\N)

C(U\)\(U\N)=

[(U\V)

u

(U\N)’I\(U\N)

(U\)’\(U\)

C(I.t(C(U\2V)))\(U\N)

CI(U)\(U\N)

[CI(U)\(U)]

U

N

Bd(U)

U

N

6

N(7")

C_

S(7")

if

(X, 7")

isN-scattered. Conversely, if

N

6

N(7"),X\N

6

,

so that

ifBd(X\N)

N

6

S(-),

then

X

isN-scatteredsince

N(7")

_C

S(7")

To see the equivalence of

(a)

with (g), note that a subset

A

C_

X

is semi-open if and only if

Bd(A)

Bd(Int(A)).

For

the remainingequivalencesit isenoughtoshow that

X

isN-scatteredif ithas anopen cover consisting of N-scatteredsubspaces. Let

{

Uo

a6A

}

be anopencoverof

X

consisting of N-scatteredsubspaces. Let

N

6

N(r).

Then each

N

CI

Ua

isnowhere dense and hence scattered in the subspace

U,,.

Also,

N

f]

Ua

is ascatteredsubsetof

X.

This shows that

N

islocallyscattered. That is, the subspace

N

hasanopencoverof scatteredsets.

It

follows that

N

isscattered,sincelocallyscatteredsets arescattered. This istrueregardless of whethersufficientseparationaxioms are presenttomake

S(7")

an deal, i.e., in anyspace,

S(7")

is alocalsubideal. Thus,

X

isN-scattered. The equivalence

(c)

:

(k)

follows from thefact that hereditarily -scattered is the same as scattered

[7].

COROLLARY

2.10.

Any

union ofopen N-scattered subspaces ofa space

X

is an N-scattered subspace ofX.

REMARK

2.11. The union ofall open N-scattered subsets ofa space

(X, 7)

is the largestopen N-scattered subset

NS(7").

Its

complement isclosed and if nonempty contains a nonemptycrowded nowhere denseset.

Moreover,

X

is N-scatteredifand onlyif

NS(T)

X.

Sincepartitionspacesare ,precisely those having no nonemptynowheredensesets,such spacesareN-scattered.

On

theother hand wehave thefollowingchainofimplications. The space

X

is discrete

=

X

is a partitionspace

::

X

is zero dimensional

:: X

is rim-scattered. Also,

X

is globally disconnected

::

X

is N-scattered.

However,

thisalsofollows quickly from theeasyto show characterization

X

is globallydisconnected

X

isan extremallydisconnecteda-space, and the factthatevery c-space is N-scattered. Actually,

something much stronger can benoted.

Every

a-spaceisN-closed-and-discrete,i.e.

N(7")

C_

(?0(7").

Of course,

CO(7")

C_

13(7")

C_

S(7"),

where

CO(7")

istheidea] ofclosed anddiscretesubsets of

(X, 7"),

and

D(T)

is thefamily ofall discrete sets. Wewillshowlater that for a non-N-scatteredspace

(X,

7")

in which

NS(7")

contains a non-discretenowheredense set, "f isnotthesmallestexpansionof

-

for which

X

is N-scattered,i.e., thereexists atopologycstrictly intermediateto7"and 7-such that

NS(cr)

THEOREM

2.12.

Let

(X

a E

A)

be a familyof spaces andlet

X

’

X

bethefree join. Then

X

isN-scattered ifand onlyifeach

X

isN-scattered.

Thefollowing

example

shows thatgenerally,evenafiniteunionofN-scatteredsubspacesfa/Isto be

N-scatteredifnotallthesubspacesareopen.

EXAMPLE

2.13.

Let

X--w

have atopology

u,

which isthe smallest expansion of the cofinite topology7"such that

7[E

C_

or, where

E

{0,

2,4,

...)

isthe set ofeven finite ordinals.

As

asubspace,

(], if[E)

(E,

TIE

isN-scatteredbeinghomeomorphicto

(X, 7-),

and

E

E u. Likewise,

JE

is

N-scatteredas asubspace being homeomorphicto

(X, 7-).

However,

JE

E

N()

and iscrowded,so that

(X,

or)

is notN-scattered though

X

(XE)

U

E

isunionof N-scattered subspaces.

We shownextthatlocal N-scatterednessisthesame asN-scatteredness but

for

theproof of that result weneed thefollowing lemm& Recall that aset

A

iscalled/3-openif itsclosure isregular closed.

LEMMA

.14. Let/beafl-opensubset ofaspace

(X,

7").

If

N

isnowhere dense in

(X, 7),

then

B

f

N

isnowheredensein thesubspace

B.

[:]

THEOREM

2.1. Ifevery pointofaspace

(X, T)

hasan N-scatteredneighborhood,then

X

itself is

N-PROOF. Let N C_

X

be nowhere dense.

Let

X

and let

U

be open

(in

J0

andN-scattered such that z

U.

By Lemma

2.14,

U

N

is nowheredense in

U.

Since

U

isN-scattered, then

U

N

is scattered andmoreoveropenin

N.

Thuseverypointof

N

has anopen scattered neighborhood.

Hence N

7 3 8 J. DONTCHEVAND D. ROSE

3.

SCATTEREDNESS VERSUS SEPARATION

In

theabsence ofseparation, a finite unionof scatteredsetsmayfailtobescattered. For example,the singleton subsetsoftwo-point indiscretespacesarescattered. Thefollowinglemmaassertsthat giventwo disjoint scatteredsubsets, if one has an openneighborhood disjoint from theother,then their union is scattered.

LEMMA

3.1.

Let

(X,

r)

beaspace and

U

E r. If

A

C_

X\U

and

B

C_

U

arescattered subsets of

X,

then

A

O

B

isscattered.

PROOF. Let

D

C_

A

U

B.

If

D

f3

B

,

then thesubspace

D

has an isolated point being scattered. Otherwise,thesubspace

D

B

D

U

has an isolated point p.

So

there exists anopensubset

V

of

X

suchthat

{p}

D

f3

(U

f3

V).

This shows that p is an isolated point of thesubspace

D.

Evidently,

A

U

B

isscattered, vI

COROLLARY

3.2.

In

every T0-space

(X, r),

finitesetsarescattered,i.e.,

I

C_

’(r).

i"3

THEOREM

3.3. Let

(X,

r)

be a non N-scatteredspace,so that

NP(r)

X\NS(r)

.

Suppose

also that

NS(r)

contains anonemptynon-discretenowhere dense subset. Then thereis atopologycrwith rCrC such that

(X,

or)

isN-scattered.

PROOF.

Recall that

NS(r)

is the union of all members of r which are N-scattered.

Let

I

N(r)P(NP(r)),

the ideal of all subsets of

NP(r),

which are r-nowhere dense. Since intersectionsof local ideals arelocal,

I

is alocal ideal. Thus,r

r*(I)

r[/]

{U\E"

U

rand

E

I

is a topologywith r

C_

rC_r

.

Moreover,

since

NS(r)

contains a nonempty non-discrete nowhere denseset

F,

it followsthat

F

is a non-discretesubsetof

(X,

r)

so that r

:/:

a.

If

(X,

r)

is N-scattered,then itwould follow also that

#

r.

In

Proposition 10of

[6],

OlavNjtadshowed that for any topologies r and r,rC_rC_r =:,r r

a.

By

Proposition 5 of the same paper by Njtstad,

N(r)

N(r).

Since

S(r)

C_

S(r),

itfollowsthat

N(r)

C._

S()

sothat

(X,

r)

isN-scattered. vI

In

searchforasmallest expansionof cr and r for which

(X,

)

isN-scattered,wehave the following improvementofTheorem 3.3.

THEOREM

3.4. Let

(X,

r)

be aspace andlet

I

{A

C_

E" E

isaperfect

(closed

and

crowded)

nowhere densesubsetof

(X,

r)

}.

Then

(X,

7)

isN-scattered,where "7

r[/],

the smallestexpanstonofr forwhichmembersof

I

areclosed.

PROOF.

It is clear that

I

is an ideal havingboth heredity and finite additivity. Further, since

I

C_

N(r),

rC_7C_r

o,

sothat as in theproof ofTheorem3.3,

N(’y)

N(r).

Thus,if

E

N(’),

as a subspace of

(X,

r),

CItE

pk(ClrE)

U

sk(CI,E),

where

sk(Cl,

E)

isthe open

(in

thesubspace

CI,

E)

scattered kernel of

CI,E,

and

pk(CITE)

E\sk(CIrE)

isthe largest perfect subspace of

CI,E.

Since r_C.y,

sk(CI,E)

is anopen scatteredsubspace of

(CI,

E,’yICI,

E).

But,

pk(ClrE)

I

=

pk(C1TE)

is discrete and hence scattered as a subspace of

(X,-y),

and is thus also scattered as a subspace of

(CI,E,

7]CIE).

It

follows from

Lemma

3.1that

CITE

isascattered subspace of

(X,

"y).

Evidently,

E

isa scatteredsubspace of

(X,

"y),

so that

(X, -)

isN-scattered. V!

It

is easy to seethatfor the topologiescrand7ofTheorem3.3and Theorem3.4respectively,"yC_

Moreover,

if anowhere denseset which is neither discrete nor contained in acrowded nowheredenseset canbefoundas a subset of

PN(r),

then "7

:#

THEOREM

3.5.

Every

closed lower densitytopological space

(X, F, I,

$)

forwhich

I

is a -ideal containing finitesubsets of

X

isN-scattered. Recall that alowerdensityspace

(X, F, I, b)

is closed if

rc

gF.

PROOF.

Recall thathere

X

is a nonempty set,

F

is afieldof subsets of

X,

i.e.,

X e F

and

F

isclosed undercomplementationand finite union,

I

isan idealcontained in

F,

and

F

F,

thelower density operator relativeto

I

satisfies thefollowing forall

A, B

E

F. Here,

A

B AAB= (A\B)U(B\A) I

(a)

(})

and

,(X)

X.

(b)

(a

.)

,(A)

(,).

ONSPACES WHOSENOWHERE DENSE SUBSETSARESCA 739

By

Corollary8.1

of[9],

I

N(’)

C’D(’)

D(-)

N(-)

C_

S(’).

Thisalso follows fromthe obviousfact that such spacesareo-spaces, r’l

Corollary8.1of

[9]

isslightlyextended asfollows.

COROLLARY

3.6. If

(X, F, I, b)

is aclosed lower density spaceand

I

isar-ideal with

I

C_

I,

then

X

S().

PROOF. By [9,

Corollary

6.2],

for closed lower densityspaces for which

I, _C

I, (X,

-)

isa

T1

crowdedspace. Theresult follows fromTheorem 2.2.

UI

COROLLARY3.7. The spaceofreal numbers

R

with the densitytopologyTd isN-scattered, and moreover, the scattered subsetsarepreciselytheLebesgue nullsets. Ui

The following theorem shows the converseof Corollary3.2 also holds andthus we have another (perhaps

new)

characterizationof

To

separation.

A

similar characterizationholds for

TI

separation.

THEOREM

3.8.

A

space

(X, r)

has

To

separation ifandonly

ifI

C_

S(r).

PROOF. Suppose

firstthat

(X,

’)

is aT0-space. Let

F

E

L.

Certainlysingletons arescattered. If

{a, b}

C_

F

witha

:f-

b,

let

NF(a)

be the smallest open subsetof thesubspace

F

which contains a.

In

particular,

NF(a)

is the intersection ofall relativeopen neighborhoods of a. Itis openbeingafinite intersectionof open sets. Likewise, let

NF(b)

be the smallestrelativeopen subsetof

F

containingb. Since

X

has

To

separation, sodoesthesubspace

F,

and eithera

JVF(b)

or b

9t

N,(a).

This proves that

F

is a union oftwodisjointproper subsets,oneofwhich isopenin

F. For

example,if b

f

NF(a),

then

F

NF(a)

U

(F\NF(a)).

Next,

weuse induotively

Lemma

3.1 onthe cardinalityof

F. Suppose

that finitesets ofcardinalityless than n are scattered. If

IFI

n, andif

F

A

U

B,

wherea and

B

are disjointproper subsets of

F,

one of which is relatively open in

F,

thenby theinduction hypothesis,

A

and

B

arescattered,andby Lemma3.1, so is

F.

Thiscompletesthe induction and shows that

I

C_

S(’).

Conversely,ifeveryfinitesubsetof

X

isscattered, choose any arbitrarypairofdistinctpointsaandb in

X.

Then

{a, b}

E

S(7")

implies thateitheraor b isisolated relativelyinthe subspace

{a, b}.

So,

thereis

anopenset

U

E

-

witha

U

and b

U,

orthereis anopenset

V

-

with b E

g

anda

t

g.

Thus,

(X,

-)

is a

T0-space.

!"1

Recall that aspace

(X,

’)

iscalleda

G2-space [3]

(originallytosatisfy condition

C2)

ifeveryinfinite subset is somewhatpreopenor equivalently ifeverynowhere dense subset is finite.

COROLLARY

3.9.

Every

C2

T0-space

isN-scattered. r-I

THEOREM

3.10.

A

space

(X,

-)

has

T1

separation ifand only

ifI

C_

D(-).

PROOF.

If

{a, b}

_C

X

witha

y

b,and if finitesubsets of

X

arediscrete,thenthere exists opensets

U

5

-

and

V

E with

U

1"3

{a, b}

{a}

and

V

13

{a, b}

{b}.

Evidently, b

U

and a

V

so that

(X,

’)

is a

Tl-space.

Conversely,if points in

X

areclosedand if

F

is afinitesubset of

X,

then for each z5

F,

F\ {}

is closedin

X,

being finite.

So,

z is an isolated point of thesubspace

F.

r’!

Thenextresultimprovesthenecessity partofTheorem 3.8.

THEOREM

3.11. If

(X, -)

isa

T0-space

andif

S

isany scattered subset of

X

and if

F

isanyfinite subsetof

X,

then

S

U

F

isscattered.

PROOF.

Theproofisbyinduction on the cardinalityof

F.

If

S

isscattered andz5

X\S,

wewill show that

S

U

{z}

is scattered. Considernowonly the

To

subspace

S

U

{z).

Let

A

C_

S

U

{z}.

If z

A,

A

isscatteredandhas an isolatedpoint. If

{z}

A,

then

A

isscattered and has an isolated point. Otherwise,z5

A

and

S

1

A

0.

Let

3/be

anisolated pointof

S

I"3

A. Let U

be anopenneighborhood of

3/such

that

U

13

S

13

A

{3/}.

Ifz

f

U,

then

U

13

A

{3/}

and

3/is

an isolatedpoint of

A. On

the other

hand,

if z

U,

then

U

13

A

{z, 3/}.

Either thereexistsan openset

W=

containingzbutnot3/,or there exists anopenset

W,

which contains

3/and

not z.

In

the firstcase,

W=

13

U

13

A

{z}

and inthe second case,

W

13

U

I"3

A

{3/}.

In

any case,

A

has an isolated point so that

S

U

{z}

is scattered.

Suppose

now that for any finite set

E

of cardinality lessthan r,

S

U

E

is scattered. Ifz5

F

and

740 1.DONTCHEV AND D. ROSE

COROLLARY

3.12.

Every T0-space

which isthe unionoftwoscattered subspacesisscattered.

PROOF.

Let

(X,

I

be

To

such that

X--Xl

UX2, where

Xl

and

X2

are scattered. Since scatterednessishereditary, wecanassumethat

X

N

X2

.

Let)

’

C

X.

Wewillshowthat

’

has anisolated point Weneedtoconsideronlythe casewhen

S

meetsboth

X

andX2,since otherwise we aredone (every subset of’a scattered spacehas an isolatedpoint).

Set

A

S

N

X]

and

B

S

N

X2.

Note

that both

A

and

B

arescattered.

Let

z be anisolatedpointof

A.

Let

U

-

be such that

U

U

isdisjoint from

B

wearedone again.

Assume

that

L

U

n

B

.

By

Theorem 3.11,

K

L

isscattered.

Let

be an isolated point of

K

andlet

V

"

suchthat

V"

N

K"

isanopenneighborhoodof" y suchthat

W

N

S

{}.

Thuseverynonemptysubset

S C_

X

failstobe crowded,i.e.

X

isscattered.

Thefollowing corollaryimproves theresult from

[1]

that

TD

separation implies

S(7)

isan ideal.

COROLLARY

3.13. Thefamily of scatteredsubsets in a

T0-space

isan ideal.

PROOF.

If

A

and

B

arescattered subsetsofa

T0-space

X,

then

A

U

B

isa

To

subspace of

X

andis a unionoftwoscatteredsubspaces.

By

Corollary 3.12,

A

U

B

isscattered. Since anysubsetof ascattered setisscattered,then in

To-spaces

the family of scattered subsetsis an ideal.

THEOREM

3.14.

A

space

(X,

T)

has

To

separationifand onlyif

S(-)

isan ideal.

EXAMPLE

3.15. Let

(X, <

be any totally ordered set. Then both the lef my and right ray topologies

L

and

R

respectively,are

To

topologies. They are not

T

if

IX[ >

1.

In

case

X

withthe usualordinalordering,

<,

L

and

R

are infact

TD

topologies,i.e.singletons arelocally closed. The space

(,

L),

where proper open subsetsarefinite,isscattered.

For

if"

:

A

C_

letnbe the least element of

A.

Then the open ray

[0,

n

+

1)

[0,

n]

intersects

A

only atn. Thus, n is an isolated point of

A.

Evidently,

$(L)

P(),

the maximum ideal.

However,

S(R)

I,

the idealoffinitesubsets.

For

if

A

isany infinitesubset of

,

A

iscrowded.

For

if rn

A

and if"

U

isanyright directedraycontaining

(U\

{m})

N

A

-y:

,

since

U

omitsonlyfinitely manypointsof.

By

Corollary3.2however, everyfinite subset is scattered. Ofcourse

S(L)

and

S(R)

are ideals inthe lasttwoexamplessinceboth

L

and

R

are

TD

and hence

To

topologies.

REMARK

3.16.

Note

that the space

(w,

R)

is acrowded

Tv-space,

which istheunionofanincreasing

(countable)

chain of scattered subsets.

In

particular, w-- U

,,{n

<

{r

<

k" k

w)

I,,

C_

S(R).

This seemstoindicatethat it isnotlikelythat an induction argument on thecardinalityofascattered set

F

can beused similar to the aboveargumentto show that

S

F

is scattered if

S

isscattered.

ACKNOWLEDGEMENT.

Research was supported partially by the Ella and

Georg

Ehmrooth FoundationatMeritaBank Ltd.,Finland.

REFERENCES

[1] DONTCHEV,

.,

GANSTER,

M. and

ROSE,

D.,

a-scattered spaces lI, Houston Journal

of

Mathematics,23

(1997),

231-246.

[2]

ELTd],

A.G.,

Decompositionof spaces,SovietMath.

DoM.,

10

(1969),

521-525.

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remarks on stronglycompactspace and semi compactspaces, Bull. Malaysia

at.

,c.,

0

() (]9s7),

[4]

ANKOVI(,

D.and

HA_MLETr, T.R., New

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(1990),

295-310.

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VAN

MK,

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and

MILLS, C.F., A

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NIASTAD,

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961-970.

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D.A.,

a-scattered spaces,

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Local ideals intopologicalspaces, preprint.

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HAMLETT,

T.I,

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