STRONG COMPACTNESS IN BITOPOLOGICAL

I

I

STRONG COMPACTNESS

IN BITOPOLOGICAL SPACES

I

I

By

T. Notnr, A. S. MesHnouR, F. H. Ksnon aNo I. A. HesnNnlN

Reprinted from lhe lndion Journol

Vol. 25, No.

l,

of Mothemolics,

lonuory 1983, Poges 33-39

r

lndion Journol of Molhemotics, Vol. 25, No.

l, Jonuory l9B3

Silver Jubilee Volume

STRONG COMPACTNESS IN BITOPOLOGICAL SPACES By

T. Notnt, A.

S MasnHouR'

F' H. KHeon AND I' A'

(Received May 2,1981; 'Revried July

l.

HeseNntN

2, 1982)

Introiluctiott

bitopologicalspaces was initiated by Kelly [1]' Recently, It/tashhour et al. [3] have introduced the notion of strongly

Tbe concept

of

compacl spaces. The purpose of the present paper is to study the concept of pairwise strong compactness in bitopological spaces' ln $ 2, we give some characterizations of pairwise strcngly ccmpact spaces' In $ 3' we define a subset pairwise strongly compact relative to a bitopological space and investigate the relationship between such sets and pairwise strongly compact subspaces. The concept of nearly compact spaces was initiated by Singal and Mathur [5]. In the last section, we introduce and characterize pairwise nearly compact spaces.

Let S be a subset of a topological space (X,'). Theclosure of S and tbe interior of S are denoted by C(S) and Int (S), respectively. A subset s is said to be preopenl2f if sc Int (c(s). Thefamily of all preopen sets in (x, r) is denoted by Po(t). A topological space (x, z) is said to be strongly compact [3] if every preopen cover of X admits a finite subcover. Let S be a subset of a bitopological space (X, 7v 'z). The closure of s and the interior of s with respect to ti are denoted by r;-Cl(S) and T ;[nt (S), respectively. However, z;-Cl(S) (resp. z;-/n I (S)) s,ill be denored by ci(s) (resp. /zr (s)) for the simplicity if the meaning is explicit. Throughout the paper, Z will stand for an index set and Vo will denote a finite subset of Z.

"r

!:r

2.

Pairwise strongly compact spqces

We begin by defining the concept

pological SJV 5

"..4::

-..

spaces.

of

strong compactness

in

bito-

34

NOIRI, MASFIHOUR, KHEDR A\ID H,ASANEIN

2.1. A family-q of subsets of a bitopological space (X, ry rr) is called rf z-pteapcn if g c PO(r) u f 0611. If, in addition, F n PO(rr) >{$) and F n PO(")=*{6}, it is called pairwise preopen (simply, DnnmrroN

p.p.). The complement of a preopen set is called preclosetl. d rrrr-preclosed family, a pairwise preclosed family, a pariwise open cover, and a pairrvise regular open cover are defined similarly. DeprNrrloN 2.2. A bitopological space (X, 11, zr) is said to be pairwise strongly compact, simplyp.s.c. (resp. M-pair\)tse srrongly compact,

simply M-p.s.c.)

if

every p.p. (resp. rrr2-preopen) cover has

a

finite

subcover.

It is obvious that pairwise strong cornpactness implies pairwise compactness. But the converse mav not be true as shown in the following

example. Let X be any infinite set, r, : the indiscrete topology and rz: the discrete topology. Then (X, z' zr) is pairwise compact but not p.s.c. since {{x}lx e X} is a p.p. covei' of X which has no flnite subcover.

The following three theorems are easily obtained from

the

definitions.

TsBonru 2. 3. A bitopological space (X, rr, rr) is M-p.s.c. r.f an,l and (X, rr) are strongly cottlpact. it is p.s.c,, (X,

only tf

"r)

Tnronru 2.4. A bitopological space(X, rr, rr) ir M-p.s.c. if antl only if every rrrr-preciosed family of subsets of X having the finite intersection property,

h,Js the nonetttply interseclion.

THnonnu

if

space (X, ,t, rr\ is p.s.c. if and only family of subsets of X having the finite inter-

2.5. A bitopological

every pairwise preclosed

section properly, Itos the nonempty intersectiort.

DrnrwtrroN 2.6. A subset ^9 of a topological space (X z) is said to be strongly compacl relative to (X, r) [3] if every cover of S by preopen sets of .(-Y, r) has a finite subcover.

Deri].rirror'i

2.7. A

subset

S

of a bitopological space (X, rr, zr) is a finite

said to be p.s.c. relative to (X,T , rr) if every p.p. cover of ,S has subcover, where the preopen sets belong to PO(r) U PO(?2).

STRONG COMPACTNESS

IN BITOPOLOGICAL SPACES

35

p's'c ' if and only if THnonBu 2.8. A bitopological space (X, t1, tr) is relative to compact rr'1 s strongly rr, f of (X, each proper ripreclose(J s'"tbset (X, rl, where i, i : 1, 2; i # i'

Proof.Necessity.LetKbeanyproperfi.preclosedsubsetolX. -K is a nonempty 7i-preopen set' L'et {U"lae Z} be a cover of K by .,r--preopen sets, then (X'-K)v{UolaeV} is a p'p' cover of X and heirce X : (X-K)U {(I..laeVo} for some 7o' Therefore' we havs relative to Kc. \) {(I.lae/r}. This shows that K is strongly compact

Then X

(X, ri).

Sttfficiency' Ler {Volaey'} be an infinite p'p' cover of Vi f6y

:

(a)

u {Zof "e

Put

{aeVlVoePO(r;)}

Then the following two cases arise

i - 1,2.

X'

Vi}

:

: X; (b) u {I/"1" e V} *

X'

Then Iirthecaseof (a) :Choose ao€Visuchthat 'L*Voo*'X' '-Vaoa exists There in u {VolaeV-i\' is a proper Tipreclosed se;t contained This shows that finite subset a; ol v.isuch that x-vorc. u {v"l"ea;1' Put {X, rr,rr) is P.s.c. In the case of (b) :

K:X-U{Vg\ae/i}' in u {Vol"eV;\' Then K is a proper ry-preclosed set contained Kc that U {V"laeA;\' Moreover' exists a finite subset zl; of h such

There we

have X

-

U

{Ir o\oe A;} c X

- K:

U {V

qlnevl}'

Thus, we obtain X

-

U

{Y

for some finite subset Ai of

,.la e A ;}

Vi

c U {V"\ae

Ai}

Consequently' we have

X: U{VoloeAiUll\. This shows that (X, z, rr) is P's'c' rr, rr) isp.s'c' if TnBonBu 2.9. A bitopological space (X, X is p's'c' relative to every proper r;preclosed subset of

i.f for i : 1,2.

and only (X' t'' r')

36

NOIRI, MASHHOUR, KHEDR AND HASANEIN

Proof. Necessily. Let Kbe a proper r;-preclosed subset of X and lU"loefl)be a p.p. cover of K, where U"ePO(rr)U PO(rr) for each oe7. Then (X-K)g {U"lae7} is a p.p. cover of X. Since (X, 11, rr) is p.s.c., we have X : t) {U"laeV}U (Y-K} for scmo V,t. This implies that Kc. g {U,laeVo}. fherefore, K is p.s.c. relative to (X, 11, rr). Sufficiency. Lettr" : {V"laer'} be an infinite p.p.covff o[X. Since either PAQt) or PO(r) contains at least two elements of {, take i and coe 7 X andVooe^{ (\PO(t;). Then tgol6:etr-i"o\\ is a. such that S *V"'t' p.p. covcr of X*yaa. Since X-tr'oo is a proper ii-preclosed subset of X, for some Vo we have X-Vooc l){V"laeVr}. Therefore, we have X : u {Voloe tro u i"0}}. Tiris sbows that (X, ry rz) is p.s.c.

3.

Subspaces of p.s.c. spaces

Let A be a subset of a topological space (X, r). The relative topologyinduced onAby ris