α Scattered spaces

(1)

Internat. J. Math. & Math. Sci. VOL. 21 NO. (1998) 41-46

41

(x-SCATTERED

SPACES

DAVID A.ROSE

DepartmentofMathematics Southeastern College of theAssembliesof God

Lakeland,Florida33801,U.S.A.

(ReceivedMarch26, 1996and in revised formJune28,1996)

ABSTRACT. a-scattered spaces, spaces whose a-topology is scattered, are introduced and used toslightlyextend a recent resultof A. V. Arhangel’skiiandP. J. Collins [1]. Strongirresolvabilityof

J. ForanandP.Liebnitz[6],isalsocharacterized intermsofthea-topology,and it isshown thatarecent theorem ofJulianDontchev[5]inessencemaximally extendstheArhangerskii-Collinsresult

KEY WORDS AND PHRASES: Scattered space, a-scattered space, submaximality, irresolvable space, strong irresolvability, heredity irresolvability.

1991AMS SUBJECT CLASSIFICATION CODES: Primary 54G15Secondary54G12,54G20.

1. INTRODUCTIONAND DEFINITIONS

For any space

X

(X, T),

the a-space ofXis

X

(X, .4)

where

{U-

NIU

E

"

and

Int

CI(N)

0}

isthea-topologyfor

X,

[9][11]. Certainly,

-

C "raandunless specifically notated,the interior and closure operatorsIntandClare withrespecttothebase topology

-.

Int andCl denote the interior and closure operators respectivelyrelativeto"r

a.

A spaceXis an a-spaceifandonlyif

X X4. The a-topology for

X

isobviouslyabase foratopologysince it isclosedunder finite inter-section,and isreallyatopologyon

X,

beingalsoclosedunderarbitraryunionof members Thislastfact canbeverified as follows. Foreachindexed subfamily

{UIA6A}

C_

,[.J(U-N)=

([.JU)-N

whereN

(I,.J.xU,)

[.,J,x(U,x

N,x).

Itisenoughtoshow thatNisnowhere denseinX. Evidently,

{UIA

A}

is an open cover for N such that for each

A,

U

NN is nowhere dense since

U

f’lN

c_

U

(U

N)

_C

N.

So, Nislocally nowhere dense Thefollowing propositionfinishes theargument.

PROPOSITION1. Eachlocallynowhere densesetsnowheredense.

PROOF. Even thoughthisresultiswellknown,asimpleargument issuppliedas we use this result againlater. Recall that asetNisnowheredenseifandonlyiffor eachnonemptyopensetUthere exists a nonemptyopen subsetVC_ Usuch thatVNN 0. SupposethatNhas anopencover

{U

A

A}

such thatforeach

A,

U

NN isnowheredense, and letUbe a nonemptyopenset IfUfN 0, set

V U. Otherwise, U

U

0

for someA

e

A. Inthiscase, setV

(U

rq

U)

CI(U

q

N)

_In eithercase,

0

7tVC_

U,

andVrqN

0.

I-I

Letussay thatafunction

f

(X, -)

---,

(Y,

a)

_{is r-continuous if}

-

_{is a}_{r-base for the weak topology}

f-(a)

{f-(V)IV

e a},

i.e., foreachopenV, Int

f-(V)

7t

0whenever

f-(V)

7t

0. Itis evident thatforanyspace

X,

theidentityfunction

f"

X

isopen and r-continuous, sothatXand X

(2)

t,2 D. AROSE

Notethat aspaceXis an a-spaceif andonlyif itsnowheredense setsareclosed Recall that a spaceissubmaximal ifandonlyif itsdense subsets areopen,whichisequivalenttohavingcodensesets closed Thisimmediately yieldsthefollowingsince nowheredensesetsarecodense

PROPOSITION2.

Every

submaximalspacetsana-space El

Recall that a space

X

is scattered ifeverynonempty subspace has anisolated point. Let

I(X)

denote the setofisolatedpoints ofX. Clearly,ifXisscatteredthen

I(X)

istheminimumdense subset of

X,

sothat dense subsetsofXhave dense interiors

PROPOSITION3. Foreach spaceX,thefollowingareequzvalent: (a) Densesubsets of

X

havedense interiors,

(b)

CodensesubsetsofXarenowhere dense,

(c) X

issubmaximal.

PROOF. Itiseasyto see that(a)holdsifand onlyif(b)holds.

Now,

if(b)holdsand

D

isadense subsetofX

,

since

X

and

X

sharedense,codense,andnowheredensesets,X-

D

is aclosed subset ofX sothatD6 r and(c)holds. Conversely,ifX is submaximal and

D

C_

X

isdense, then

D

E"r

so that

D

U N for some UEr and nowhere dense N Now, U

Cl(N)

C_

Int(D)

implies

X

Cl(D)

CI(U)

c_

Cl

Int(D)

sothat dense subsetsof

X

havedense interiors El

An

immediateconsequenceisthatforeachspace

X,

if

X

isscattered thenX issubmaximal,andat oncewehave the following resultof A. V.Arhangel’skii and P.J.Collins ].

COROLLARY1.

fix

isscattered, then

X

if

andonly

if

X

issubmaximal.

PROBLEM. Findthelargestclass ofspacesforXsupportingthe conclusionof Corollary

Itisclear that we arelooking forthe largestclass (ifitexists) of spaces forXso that

X

implies

X

is submaximal. Enlarging the class ofscattered spaces to contain the non-a-spaces only trivially extendstheArhangel’skii-Collins result. Perhapsanontrivialstrengthening of Corollary results by assuming only that

X

isscattered. Certainly,the conclusionwould besupported.

2. a-SCATTERED SPACES

DEFINITION1.

A

space

X

isana-scatteredspace

if

X

isscatterec Somecharacterizationsof a-scatteredspacesarestatedinthe following theorem.

THEOREM1. Foraspace X,thefollowingareequivalent:

(a) X

isscattered,

(b) Everysomewhere densesubspaceof

X

hasan isolatedpoint.

(c)

I(X)

isdensein

X.

PROOF. Since

X

and

X

share the same somewhere dense subsets and sincerC_

,

(b)

implies

that each somewhere densesubspace of

X

hasan isolatedpoint. Consequently, (a)follows from(b)

sincethe nowhere densesubspacesof

X

are discrete. Tosee that(a)impliesCo),letAbe anonempty subspaceof

X

havingno isolatedpoint. Wewillshowthat

A

isnowhere dense. AssumingthatX is scattered,

I(A) #

0where

I(A)

denotes thesetofisolatedpointsinthesubspace

(A, TIA)

Foreach p

I(A),

there exists

U

N T (withtheunderstandingthatU randInt

Cl(N)

))

such that

(U-

N) A

{p}.

Since

I(A)

,

p6

CI(N).

Thus,

U

I(A)

c_

Uq

A

CNt3

{p}

C_

Cl(N

), anowhere dense set,implies

U

f31

(A)

is anowhere dense subset ofX. Thisshows that

I

(A)

isa locally nowhere dense subset of

X

andhencebyProposition1,

I

(A)

isnowhere dense. It followsthat

A

I(A)

(X-

I(A))

t

A

6

"r[A.

Thus,

I(A

I(A))

0 implies

A

I(A),

so that

A

is nowheredense.

Clearly,Co)implies(c)sincenonemptyopensetsaresomewheredense and foreachopensubspace

U

of

X,

I(U)

C_

I(X).

To show that

(c)

implies (b), let

A

c_

X

be somewhere dense and choose

p

I(X)InCl(A).

Then

{p}

r and p

Cl(A)

implies p6A. Finally, this implies that

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a-SCATI’ERED SPACES 43

Sincethe a-scattered spacesare precisely those havingadensesetofisolatedpoints,examplesof suchspacescaneasilybefoundwhich are notscattered. Onesuchexampleisoffered.

EXAMPLE 1. Let

(X,

)

be the set

of

realnumbers wth the smallestexpanszon

of

the usual

topology

for

whichrational points areopen. Then

I(X)

Qisdense so that

X

isa-scatterec But, thesetP X Q

of

irrationals zs

dense-zn-itself

so thatI

(P)

@andXtsnotscattered.

Thefollowingtheoremdecomposes scatterednessintotwostrictlyweaker components

THEOREM2. ,4_spaceXisscattered

if

andonly

if

Xisa-scattered andevery nonemptynowhere densesubspace

of

X

hasanisolatedpomt. [2

REMARK. Julian Dontchev hassuggestedthatTheorem2might berephrased morenaturallyas follows. A space

X

is scattered if and only if

X

is a-scattered and N-scattered whereaspaceis

N-scatteredifevery nowheredense subset isscattered.

in-itselfandyetnowhere densesubspacesofX are discrete.

Wenowoffer the following slight improvement of the Arhangerskii-Collins result.

THEOREM3.

If

X

isa-scattered, then

X

issubmaximal

tf

andonly

if

X

tsana-space. [2 3. STRONGLY IRRESOLVABLE SPACES

Letuscall aspace crowdedif it is dense-in-itself. Edwin Hewitt in

[8]

showed theexistence of crowdedsubmaximalspaces of arbitraryinfinitecardinality. Suchaspace

X

isofcourse ana-space but isfarfrom being a-scatteredsince

I(X)

@. Canthe Arhangel’skii-Collins result be extendedto aclass of spacesnotrequiring theexistenceofisolatedpoints? Yes Julian Dontchev lifted theA-Cresulttothe class of strongly irresolvable spaces in [5] Recall that a space is strongly irresolvable [6] ifeach nonempty open subset is irresolvable.

A

space is irresolvable, i.e., not resolvable, if it cannot be expressedastheunionoftwodisjoint dense(or codense)subsets. Westatethe Dontchev result formally and thenshowthatinessence,itgivesthebest possibleextensionof the class of spaces supporting the

A-Cresult.

THEOREM4. (Dontchev

[5])

If

X

sstronglyirresolvable, then

X

issubmaximal

if

andonly

tf

Xisana-space.

Actually, Theorem4is acorollary ofthefollowingobservation.

TliEOREM 5.

A

space

X

isstronglyirresolvable

if

andonly

if

X

issubmaximal.

PROOF. ByProposition 3,X is submaximal ifandonlyifevery codense subset of

X

isnowhere dense. Suchaspace

X

mustbestronglyirresolvable. Otherwise, thereis anonemptyopensetUwhich is a union oftwo disjoint nonempty codense subsets. But sincecodense subsets ofanopen set are codensein

X,

andhence nowhere dense,thisforcesthecontradictionthatUisnowhere dense.

Conversely, if a nonempty space X is strongly irresolvable and if

D

C_

X

is a dense subset,

Int(D)

sinceXisirresolvable. Further,

Int(D)

isdense for otherwise,

D

Cl

Int(D)

wouldbe a

denseandcodensesetintheopensubspace

X-

Cl

Int(D).

Certainly, if

U

c_

X-

Cl

Int(D)

is open,UE%thetopologyon

X,

implies

U

n

D

,

and alsoU

D

#

@

since

U

N

Int(D)

.

[2

Adecomposition of submaximalitynowfollows.

COROLLARY 2. A space

X

issubmaximal

if

and only

if

X is strongly irresolvable and

X=X

.

[2

It is easytoproduce irresolvablespaces whichare not strongly irresolvable, showing that strong irresolvability is strictly stronger than irresolvability. For example, if

Y

is any noncmpty irresolvable space and

Z

isanynonempty resolvablespace,the free join

X

Y

LI

Z

cannotbestronglyirresolvable.

(4)

/,2, DA.ROSE

be hereditarily irresolvable since the subspace

P

ofirrationals is resolvable. The countable set of irrationalalgebraic numbersand itscomplementthe setof transcendentalnumbersforma resolutionof

P

as a disjoint union oftwo dense subsets. Evidently, strong irresoivability lies strictly between the properties of irresolvability and hereditary irresolvability Of course, strong irresolvability is strictly weaker than submaximality since submaximalityimplies hereditary irresolvability. To see that strong irresolvability is also strictly weaker than the cz-scattered condition, consider the crowded infinite submaximal spaces ofHewitt. Having no isolated points, theycannot be a-scattered, and yet being submaximala-spaces, theyarestronglyirresolvable.

Wenow want to note some similarities inbehavior for submaximality, hereditary irresolvability, and scatteredness in contrast to corresponding behaviors for a-scatteredness and strong irresolvability Recall thatapropertyissemitopological[3],if it ispreservedby semihomeomorphisms, bijectionswhich arebothirresolute andhaveanirresoluteinverse, i.e., bijections forwhichimages andinverseimages of semiopensets aresemiopen. A set

A

issemiopen if and onlyif

A

C_ Ul

Int(A)

Itwas later shown implicitlyin

[2]

and later independently and explicitlyin[7]that semitopological propertiesareprecisely the a-topological properties. The same result had essentially been stated by O. Njstad

[12]

as a consequence ofthe fact observed in a former paper [11], that in any topological space, the classof semiopensets determinedthea-topologyand vice-versa. Recall that apropertyPisa-topologicalif both

X

and

X

have

P

when eitherhas

P (see

also

[14]).

Example above shows that scatteredness, submaximality, and hereditary irresolvabilityare notsemitopological properties. Itwasnotedin

13]

that crowdednessissemitopological.

Moreover,

havingisolatedpointsissemitopologicalsince one canshow that for any space

X, I(X)= I(X).

In fact, since X and X share the same dense sets, a-scatterednessissemitopological. Also,strongirresolvabilityissemitopological foraspace

X

isstrongly

irresolvable if and only if

X

(X)

is submaximal which occurs if and only ifX is strongly

irresolvable. As a side remark, it was noted in [7] that resolvability is semitopological. Thus, irresolvabilityisalso semitopological.

For a further comparison-contrast, unlike scatteredness, submaximality, and hereditary irresolvability, evidently, a-scatteredness, strong irresolvability, and irresolvability are not hereditary properties. Hereditary a-scatterednessisscatteredness and hereditary strong irresolvabilityishereditary irresolvability whichisstrictly weaker than submaximality. Perhaps,theproperty strong irresolvability could be renamed a-submaximality in light ofTheorem 5. However, note that in this sense, a-submaximality, a-hereditary irresolvability, and a-strong irresolvabilityarepair,vise equivalent.

It might also be observed that unlike scatteredness, submaximality, hereditary irresolvability, and irresolvability, the a-scatteredness and strong irresolvabilityproperties are not generally preservedby

opensurjections.

EXAMPLE

2. Let

(Y, )

be any

infinite

a-scattered space andlet

Z

be any countably

infimte

set disjoint

from

Y

endowedwith the

cofinite

topology. Let X

Y

U

Z

be thespace wth topology

"

crU

{

UC_X

X

Uis a

finite

subset

of

Z

}

andlet

Y

Z

bethe

free

join

of

Y

andZ. Then the identity

function

f"

X

-

Y

U

Z

is an open surjection

from

the a-scattered and hence strongly

irresolvablespace

X

onto aspacehavinganonempty resolvableopensubspaceZ. Hence,

Y

Z

is neitherstronglyirresolvable nora-scattered

REMARK. Thespace XofExample2has

To

separationbut failstobea

T

spacesince points of

Y

are notclosed. In fact,thespace

X

of theexamplemay bereplacedbya

TI

regular spacewithout disturbingthevalidity of theexample. Let

Y

Q

be thesetof rational real numbers andlet

T

be theset ofrationalnumbers having terminating decimalexpansions. Then withrespecttothe usual subspace

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a-SCATTERED SPACES 45

B,(z)

([(y

r, y

+

r)

n

Q]

1}) u

[(y

+

r] r, y

+ +

r)

Q]

beabasicopenneighborhood ofz

Foreach y Y

T,

md each positive ration numberr, letB(y) (y r,y

+

r)

Qbeabasicopen

neighborhood ofy. Finely, ify

T,

let

{y}

E Since y

+

Bis iffationforchy Q,itfollowsthat

(X, w)

is

T

md

rel.

so,

Xisa-scaeredsince

Y

is open d a-scaered subspacemd2is nowhere dense. The

identi

map

f

X

Y

UZism

on

bijectionyet the

T

md

rl

&ee

join

YU

Z

isnotronglyiesolvable since theopen subspaceZis

homeomohic

to(Q,

v)

However, the properties a-scaeredness d strong

iesoivabili

e preseed by open -cominuous

suoections.

Inpmicul, if aproduct spaceis either scaRered, herditmlysolvable, submm,

a-scarerS,

or

strony

iesolvabl, then soiseach favor Ontheotherhd, xple

is

ven

[10]

of a subm space X whose sque is not submm so that generly,

submm

is notproduive.

,

(a-)scaeredness

isnotitely productivesince acoumly

itpower ofatwopoimdiscretespaceis

homeomoc

to

e

usu

crowdedd

compa

Cmtor s. Ts

so

shows that

song

ielvabili

is notitely produive

sce

E H

[8],showed that locly

compa

crowded Hausdospacese revivable.

However,

eve

fite

pru

of

a-scaRer

spacesis

a-scatter.

Forif

X

md

Y

espacesth densesetsof isolat poims

I(X)

d

I(Y)

resptively, then

I(X

x

Y)=

I(X)x I(Y)

is dense in XxY. Is

scattereess

fitely productive? Is

grong

iesolvabili

fitely productive? Both of these questions e mswered

afively

in asequeltots paper [4].

ACOWLEDGMENT. Theauthorwould liketothchoftherefersforhelpl

suestions

wchlto vuleimprovemems oftspaper.

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PJ.,

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