LOCAL DISCRETE EXPANSION

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 4 (1996) 747-750

747

APPLICATION

ON

LOCAL DISCRETE EXPANSION

M.E.ABD EL-MONSEF, A.M. KOZAEandA.A. ABOKHADRA DepartmentofMathematics,Faculty ofScience

Tanta University,Tanta, EGYPT

(Received June2, 1992andin revisedform September 14, 1995)

ABSTRACT. The processofchanging^atopologybysome types of itslocaldiscreteexpansion preserves s-closeness, S-closeness, semi-compactness, semi-T, semi-R, E

{0,1,2},

^{and extremely dis-}

connectness Via some otherforms of such above replacements one can havetopologies which satisfy separationaxiomsthe originaltopologydoesnothave

KEY WORDS AND PHRASES: Near open sets, local discrete expansion, extremely disconnected, semi-compact,s-closed,S-closed, semi-T, semi-/, andcidspaces

1991AMSSUBJECTCLASSIFICATION CODES: 54A10, 54D10, 54D30,54G20 1. INTRODUCTION

Throughout the present paper

(X, 7-)

is a topological space (or simply a space X) on which no separation axioms areassumedunless explicitlystated. For any BC

X, clTB

(resp int7 B)denotesthe closure (resp interior) ^of/3 A subset Bis said tobe regular open (resp regular closed)ifB int,

(clT(B))

(resp B

-clT (int(B)))

^{A subset B of}aspaceX is saidtobe r-semi open [12] (resp ^7-- regular semi-open[2]) ifthere existsar-open(resp. r-regular open)setUsatisfyingUc/3

c clTU B

is r-semi-closed

[3]

^ifthesetX-

B

isr-semi-open. Thefamily of all regular open (resp regularsemi- open, semi-open) ^sets in

X

is denoted by

RO(X, 7-)

(resp

RSO(X, 7-),SO(X, 7-))

The union (resp intersection)^{of all}7--semi-open(resp r-semi-closed)sets contained inB (resp containing/3)iscalledthe 7--semi-interior [3] (resp 7--semi-closure[3])of

B,

and it isdenotedass-intB (resp ^s cluB) Aspace Xis saidtobeextremelydisconnected(denoted by E.D ifforeveryopensetUof

X, clU

^isopenin7-

Theconcept of localdiscreteexpansion ofatopologywas firstintroducedby S P

Young

in 1977 [17],

"Let

(X, 7-)

be a topological space and

A

be any subset of X The topology

7-[A] {U- ^{H U}

E7-,

H c A}

^iscalled the localdiscreteexpansion of7-by

A

A space

X

issemi-

T2

^{[13] (resp}

semi-T ^[1])

iff for x, y E

X,

^x

:/:

y thereexist U and V

SO(X, 7-),x

Uand y V such that U

n

V (resp

cl.U

fqclV---).

Semi-T0

^and

semi-T1

^{were introduced to} topological spaces[13]by replacing the word

"open"

by"semi-open" inthedefinitions

ofT0

^and

T1

respectively A space

X

is

semi-R0

[6]iffforeach semi-opensetUand x E

U,

s

cl {x} ^c

^{U AspaceXis}

semi-R1 [6]

^ifffor x, yE

X

such that s-

clT{x} :

^s-

^cl{y}

^thereexistdisjoint semi-opensets Uand Vsuch that s-

clT{x} c U,

and s-

cl.{y}

C V. A space

X

is called cid [15] ifevery countable infinite subspace ofXis discrete. A space

X

issemi-compact

[7]

(resp s-closed[5],^S-closed

[16])

^iffor every cover

{V,:i I}

^{of X} by semi-open sets of

X,

there exists a finite subset

I0

^{of I such that} X t2

{V

E

I0}

(resp

X

t2

scl(V,):

E

Io},X

^t2

cl(V):

E

I0}).

REMARK

1.1. For a subset

A

of a space

(X, 7-)

we say that

A

satisfies condition

(C1)

^if

A

^t_JU

,

forevery

U

^7-

{X}.

Listedbelowaretheorems that will be utilizedinthispaper

THEOREM 1.1

[14]

^{If7- and 7-’} are two topologies on

X

such that 7-

c 7-’,

then

RO(X, T) RO(X, 7-’) iffclG

^{cl,,G forevery G 7-’} [equivalent iffint,

F

int-,F, for every

F

THEOREM 1.2

[11]

^If

X

is aspace, and

A c X

satisfying

(C1)

Then,

climlG

^{cloG, for every}

GE

7-[A]

7148 M I. ABI)I’;I.-MONSEF, AM KOZAF.andAA ABO KtlADRA

THEOREM 1.3 [4] If

X

is a space, and AE

SO(X,7-)

such that

A

C

B c

cl,A

Then.

B SO(X, 7-)

THEOREM 1.4[10] IfXisaspace,andBC X,thens cl,B BUint,cl,B

THEOREM 1.5 [8] AspaceX isE D iffforeverypairUandV ofdisjoint7--opensets, wehave clU ^cl,V

THEOREM1.6[5] AspaceXiss-closed iffeverycoverofXby regular semi-open^{setshasafinite} subcover

THEOREM1.7 15]

(a)

AspaceXis cid ifeverycountable infinite subsetisclosed (b)

Any

infinite cidspaceis

T

THEOREM1.$ 17] Let Abe any subsetofX Then

(A, 7.[A] A)

is discrete

TItEOREM1.9 17] Let

A

be aclosed subset of

X

Then

(A,

^7-

A)

is adiscretesubspaceofXff r

-[A]

THEOREM 1.10

[9]

Let X be a

T-space

Then X is cid iffcountable subsets have nolimits points

2. ON LOCAL DISCRETE EXPANSION

THEOREM2.1. If

(X, "r)

isaspaceand

A c X,

then (i)

SO(X, 7-[A])

^C

{t3-

H:13

80(X,7.),H

C

A}

(ii) ^IfAsatisfying

(C),

then theinclusionsymbolin(i)isreplaced byequality sign

PROOF. (i) Let W

SO(X,-r[A]),

^then there exists V

7.[A]

^{such that V C} W

c CLIAI

^V

Then

(U H) c

W

c cl-rIal(U- ^H),

^where

^U

^7-,

H

^C

^A

^Put

H

^U

^H,

^then

H2

^C

^A,

and

(U- H1)U H ^c

^WU

H

^C

clr[Al(U- H)U H

^{Then U}

^c

^WU

H ^c cl.,-[AIU

^Ccl-U,

and

(WUH2)SO(X, 7-)

Put B=WUH2, and

H=H1-WcA

^{Then B-H=}

WU(UfqH1)-(HI-W) =W.

(ii) ByTheorem 1.2,theproofis obvious

REMARK2.1. FromTheorem2.l,it iseasy^toprovethat,for any

A

C X

SO(X, T)

^c

SO(X, T[A])

TIIEOREM2.2. If

(X, 7-)

is aspace,and

A c

Xsatisfying

(C1)

^Then"

(i)

SO(X, 7-) SO(X, T[A]).

(ii)

RSO(X, T) RSO(X, T[A]).

PROOF. In general

SO(X, 7-) c SO(X, T[A]).

^To prove the converse, let W

SO(X, T[A]),

then there exists VE

"r[A]

satisfying

V c

WC

cl.[AlV.

^Then

(U H)

C

W c cllAl(U ^H),

U_ET,H CA. Thereare two cases.

(a) U X,

thenU H U Since

cltAlU

^cl,

^U,

^thenW

^{SO(X, 7-).}

(b) U=X,

^then

(X-H)

CWCcl,

tAI(X-H) Ccl,(X-H).

^{Since AfqU=, then}

cl-A c (X U),

^{and cl,AO}U

,

^{implies to cl,}

H

^fqU

,

for each U

e

T

{X}

Hence U cl,

H,

and int.cl,

H ,

^and

^H

is a T-semi-closed set Thus

(X-H) SO(X,T)

^From

Theorem 3,W

SO(X, T)

(ii) ByTheorems1.1 and 2, theproofisobvious

COROLLARY2.1. IfXis aspace, and

A c X

satisfying

(C1)

^Then (i)

(X, r)

is

semi-T

^iff

(X, 7.[A])

^issemi-T,(i

{0,1, 2}).

(ii)^If

(X, r)

^is

semi-T,

^then

(X, r[A])

^is

semi-T.

(iii) If

(X, r)

^issemi-R, then

(X, r[A])

^is

semi-R

(i

{0, 1})

PROOF. ByTheorem(2 2),theproofis obvious

TIIEOREM 2.3. IfXis aspace, and

A c

Xsatisfying

(C1).

Thens

cl,-[A]G

^s cl.,-G, for everyG

T[A]

PROOF. LetG

7.[A],

^{then s}

clqA]G

GU

int-[a] cl.,-[A]G

^G

int.,-cl.,-[A]G

^{G intcl.,-G}

cl-G

[byTheorems 1,12and14]

APPLICATIONONLOCALDISCRETE EXPANSION 749

THEOREM2.4. If

X

is aspace,andA

c

Xsatisfying

(C1).

Then

(X, 7.)

^{isE.D. iff}

(X, 7.[A])

^{isE D} PROOF. Let

(X, 7.)

be

E.D.,

WE

7.[A]

^{ThenW U}

H,

U E

-,

H

c

^A.

But

cl.[A](U ^H) cl.[Ai ^U

^{clU, and}

cl.U

^{7.. Thus}

cIT[A]W 7.[A],

^and

(X, 7.[A])

isE.D Conversely, let

(X,7.[A])

^be

E.D.,

and

U,V

7. such that

clTU

^fqcl.V

=/= . ^By

^{Theorem 1.2,}

c/[A]U

^N

cl-[A]

^V

,

^thenUNV [byTheorem

1.5].

^Hence

(X, 7.)

isE.D.

THEOREM2.5. If

X

is aspace,^and

A c X

satisfying

(C1).

^Then

(X, 7.)

^issemi-compact(resp s-

closed)iff

(X, 7.[A])

issemi-compact(resp. z-closed).

PROOF.

By

Theorem 2.2, theproofis obvious.

THEOREM2.6. If

X

is aspace,and

A c X,

and

(X, 7-[A])

^isS-closed(resp.z-closed),then

(X, 7.)

isS-closed(resp. z-closed).

PROOF. Since

SO(X, 7.) ^c SO(X, 7.[A]),

^{theproofisobvious.}

3. L-

Ti

^AND

Q

L-

Ti

^SPACES

Let

R

beatopological propertywhich ispreservedunder expansions

DEFINITION3.1.

A

topological space

(X, 7-)

iscalled

L- R

ifthereexists asubsetS

c

X and S

X,

such that

(X, 7-[S])

^hasR.

PROPOSITION3.1. IfT-

c -r’,

^{then for}any SC

X, 7-[S] ^c "F[S].

REMARK

3.1. If7-C

T’

and 7- is

L R,

then 7-’ is also L

R,

e. any expansion ofL

R

topologyon

X

isalsoL R.

DEFINITION3.2. Let 1, 2, 2.5 andj 0, 1, 2, 2.5. We say that

(X, 7-)

is

Q L T,,

if it is

L T,

and

T

^wherej

^<

^i.

Nowwe aregoingtoshowthat some of the properties

L T,

andQ

L T,

are satisfied forsome spacesbutnotforsomeotherspaces.

PROPOSITION3.2. Foraspace

X,

the following diagramiseasilyobtained.

T2 = ^{Q L-} _T2

^=:>

^T2 = ^{Q L- T2 => T1}

=>Q-L-T1

=:>To.

EXAMPLE

3.1. Let

X- {a,b,c,d}

^andT-

{,X, {a,b}, {c,d}}

^{is not}

To

^if

^{A- {a,c},}

^then

7-[A] {,X,{b},{d},{b,d},{a,b},{c,d},{b,c,d},{a,b,d}} isTo.

^{Thisexampleis}

Q- L- To.

Thefollowingis anexample^{of a}Q L

T2.

^butnot

T2..

EXAMPLE

3.2. Let

X

Nx

Z

U

{

^1,

0),

1,

1)}

^{whereNisthenaturalnumbersandZ}

the integers. Thetopologyhasas itsbasesetsof the following forms:

{(re, n)},

n0,

m-

^-1

u,,((,,o)) {(,,o)} u {(,,,,)[ ^{I,1 > ,,,},}

U,.,((-1,1))={(-1,1)}u{(,mlla_>..,m>O ^},

^heN

U,.,((-1,-1))={(-1,-1)}U{(a,m)[a>_n,m<O},

^ne.N.

Thisspaceis

T2

^butnot

T..5

^as

^1,1)

^{and 1,}

¹⁾

^{donot have}disjoint closedneighborhoods.

Choosing

A

Nx

(Z {0}),

^thediscreteexpansionisthediscretetopologyand thus

T2.

EXAMPLE&&

LetX {a,b,c,d}

and ^T=

{,X,{b},{d},{b,d},{a,b},{c,d},{a,b,d}, {b,

c,

d} },

^then

T[A]

^{Discrete. Thisexampleis}

^{Q L-} Tl

^{but not}

T1

^{andisanexample ofaspace}

which isnot

Q L T2.

EXAMPLE

3.4. Let

X {a,b,c}

^andT

{,X, {a,b}}.

^If

^A {a,b},

^then

7-[A]

^Discrete

Thisexampleis not

Q L T1.

The excluded pointtopologyon an infinite setXisthe family consisting of andall subsets ofXnot

containingapointp ofX.

EXAMPLE

3.5. The excluded point topologyis

L- T1

^{and not}

^L- T2 (also

^{is an} example of Q

L T1

^butnot

^T1).

PROOF. If

X

is an infinite setand pistheexcludedpoint and

A

C

X,

then:

(i) Ifp

A,

wehave

7-[A]

^TU

{X B

BC

A}.

^Thus

7-[A]

^is

T1

^butnot

T2.

750 M1.. AiI)I:I.-MONSI.I:,AM ^K()ZAI.:andA A ^AI]()KIIAI)RA

(ii) IfpE A,thenAisclosed,andtherearetwocases

(a) IfBc A,p E Bin this caseany opensetin

-[A]

isopenin7., e 7.

-[A]

(b) IfBc A,pBas(i)

Thus7.[A]-7-L{X-B-Bc A}

EXAMPLE3.6. Let

X-[0,1]

^and

7.={q,X, AcX.X-A

is

finite}

^IfwetakeS=

(0,1],

then

"r[S]

^isthe Discretespace ThisexampleisQ L T_,butnot

T.,,

THEOREM3.1.

(X,

7.) is cidspaceifft.

7.[A]

^whenever

A

is acountableinfinitesubset ofX PROOF. Weassumethat

(X,

^7.) iscid, then

A

isclosedanddiscretesubspace

By

Theorem 9 we have that7-

"r[A]

^Converselywe assumethat"r

"r[A]

^ByTheorem 8,wehave that

(A,

^"rFI

A)

^{is a}

discretesubspace ofXand

(X,

7-)is cidspace TtIEOREM3.2. Everyspace

(X,

7.) isL

To

PROOF. Assume that

zo

^{X We aim to}prove that

7-IX- {To}]

^is

To

^{For this purpose let}

z,t X,x /,ifU G"ris an opensetcontainingz, thenU-

{/}

is anopensetin

"r[X- {zo}]

andnot

containing Ifx0 :r, thenX-

{//}

is an openin

7.[X- {x0}]

^andnotcontaining_V Thiscompletes theproof

Thefollowingexampleillustrates a

Q L

space butnot

T2

EXAMPLE 3.7. (Countablecomplement topology[16]) IfXis an uncountable set, we definethe topology of countable complementson Xby declaring open all setswhosecomplements arecountable, together with $ and X

(X,

7-) is

T

^{but not}

T2

^{Let AC X such that}

X-A

is countable For

x0 X

A,

AtO

{To}

^{is T-open, and so}

(A

^tO

{:r0})

^A

{To}

E

r[A]

For

a:o A, A

isr-open, which meansthatA

(A {To}) {a:0}

^is

r[A]-open

Thus

r[A]

is discreteand consequently

T2

UNSOLVED PROBLEM. If

(X, "r)

is a spacewhich does nothave a property

P,

what are the properties of the subset

A

that make

(X, r[A])

^haveP (for P fixedproperty)

ACKNOWLEDGMENT. Wewould like tothank the referee for valuablecommentsand suggestions, especially Example32,Example37andTheorem32

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