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sets and systems Fuzzy Sets and Systems 98 (1998) 207-210

On fuzzy almost compact spaces M . N . M u k h e r j e e * , R.P. C h a k r a b o r t y Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Calcutta 700 019, India

Received February 1996; accepted October 1997

Abstract

In this article, an investigation of almost compactness for fuzzy topological spaces including certain new characterizations is attempted in terms of fuzzy OC-sets, ~-open sets, ~-closure operator, O-open sets and Co-accumulation points. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy almost compact space; Fuzzy O-closure; Fuzzy O¢-set; Fuzzy a-open set

The notion of almost compactness for topological spaces was introduced by Alexandroff and Urysohn [1] in 1924 while the study of the same concept for fuzzy spaces was initiated by DiConcilio and Gerla [5]. Although the investigations in the latter case have not been intensive enough so as to exhibit all the important applications and elegance as those found for the former case, attempts are on to tackle the notion of fuzzy almost compact spaces through various approaches. In this connection the investigations in [3, 8, 13] are worth mentioning. The present note is yet another attempt to the study of fuzzy almost compactness via some new approaches. Throughout the paper, by (X, T) or simply by X we shall mean a fuzzy topological space (fts) as defined by Chang [4]. According to the notation of Pu and Liu [12] a fuzzy point with the singleton support at x and value t (0 < t ~< 1) at x will be denoted by xt. The constant fuzzy sets taking, respectively, val* Corresponding author.

ues 0 and 1 at every point of X are denoted by 0x and lx, respectively. For a fuzzy set A in X, (1 - A ) will stand for the complement of A. Two fuzzy sets A and B are said to be q-coincident [12], to be denoted by A q B, if there is some x E X such that A ( x ) + B ( x ) > 1. The negation of such a statement is written as A ~ B. A fuzzy set U in X is called a q-neighbourhood (qnbd) of a fuzzy point xt if there exists a fuzzy open set V with xt q V <~U [12]. By clA and intA we shall denote, respectively, the closure and interior of a fuzzy set A in an fts X. A fuzzy point xt is called a fuzzy O-cluster point of a fuzzy set A if cl V q A holds for every open q-nbd V ofxt [11]. The union of all fuzzy O-cluster points of A is called the fuzzy O-closure of A [11], written as O-clA; and A is called fuzzy Oclosed ifA = O-clA [11]. The complements of fuzzy O-closed sets are called fuzzy O-open [11]. The fuzzy O-interior of a fuzzy set A in X, written as O-intA, is defined to be the fuzzy set 1-O-el(1 - A) [6]. It has been shown in [6] that a fuzzy set A is fuzzy O-open iff A = O-intA. The following result of [10] will be used in the sequel.

0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PII S01 65-01 14(97 )003 60-6

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Lemma 1. For a fuzzy open set A in an fts X, clA = O-clA, and dually, for a fuzzy closed set A, int A = O-int A. We recall the following definitions. Definition 2 (DiConcilio and Gerla [5]). A family q/ of fuzzy (open) sets in an fts (X, T) is called a fuzzy (open) cover of X if sup {U(x): U E q/} -- 1, for each x E X. The fts (X, T) is said to be fuzzy almost compact if every fuzzy open cover q/ of X has a finite proximate subcover, i.e. there is a finite subfamily q/o of q/such that U{cl v: V E q/o} -- Ix. Definition 3 (Chakraborty and Malaker [3]). A fuzzy set A in an fts X is said to be a fuzzy O~-set if A _ O-cl B, for some fuzzy set B in X. Remark 4. It is easy to see that for any fuzzy set A in an fts X, O-intA (O-clA) is fuzzy open (resp. fuzzy closed). Hence every fuzzy OC-set Js fuzzy closed. In the next theorem we obtain a characterization of a fuzzy almost compact space in terms of a sort of fuzzy cover by fuzzy O~-sets. Theorem 5. An fts X is fuzzy almost compact ifffor

every fuzzy cover q~ of X by fuzzy OC-sets, with the property that for each fuzzy point x~, some members of q~ is a q-nbd of x~, has a finite subfamily which is again a fuzzy cover of X. Proof. Let X be a fuzzy almost compact space and q/ be a fuzzy cover of X with the stated property. Suppose N denotes the set of natural numbers. Now, for each n E N and each x EX, consider the fuzzy point xl/n. Then, there are some U~ E q/and a fuzzy open set Ox" such that xl/n qO n <~Ux~. This gives Ox~(X)> 1 - 1/n. Thus, {on: n E N and x E X } is a fuzzy open cover of X. By fuzzy almost compactness of X, there exists a finite subset {xl,x2 . . . . . xm} of X and finitely many positive integers nx,...,nm such that

lx=OclOni<~OclU~i-~Ofxn/ i=1

i=1

i=1

(as each U~', being a fuzzy OC-set, is fuzzy closed). Thus, {Ugh', U~"~. . . . . un; ' } is a finite sub-

collection of q/, which is also a fuzzy cover of X. Conversely, let the given condition hold for an fts X, and let q / b e any fuzzy open cover of X. Then by Lemma 1, Y/"= {cl U: U E q/} is a fuzzy cover of X by fuzzy OC-sets. Also, for any fuzzy point x~, there is Ux~ Eq/ such that U x , ( x ) > l - a (as sup {U(x): U E q / } = 1, for each x E X ) . Thus, Ux~ is a fuzzy open set in X with x~ q Ux~, which shows that cl Ux~ is a q-nbd ofx~. It then follows by hypothesis that there is a finite subset {xl ..... xk} of X such that lx = Uf=l cl Ux~,proving that X is fuzzy almost compact. [] Let us now recollect that a fuzzy set A in an fts X is called fuzzy a-open [7] (semi-open) [2] if A~

MN. Mukherjee, R.P. ChakrabortylFuzzy Sets and Systems 98 (1998) 207-210

209

Now, int clint V is a fuzzy open set such that xtq V~intclint V (as V is fuzzy ~-open). Thus xt ~ clA. Consequently, cl A ~ ~-cl A. Hence ~-clA = c l A . []

V in X such that x~ q V <~cl V ~< U. Thus, for each fuzzy point x~ ~ 1 - U, there is a fuzzy open set V with x~ q V such that el Vq(1 - U). Hence 1 - U is fuzzy O-closed and hence U is fuzzy O-open. []

Remark 9. Since every fuzzy ~-open set is fuzzy semi-open, the above lemma is equally valid, whenever A is any fuzzy ~-open or in particular, a fuzzy open set.

Theorem 13. In a fuzzy almost compact space X, every fuzzy cover of X by fuzzy O-open sets has a finite subcover.

Theorem 10. An fts X is fuzzy almost compact iff every fuzzy cover q~ of X by fuzzy ~open sets has a finite subcollection q/o such that U{~-cl u : u E q/0} = lx. Proof. Let X be fuzzy almost compact and q/ be a fuzzy cover of X by fuzzy c~-open sets. For each U E q/, put int U = U °. Then U ° ~

(1)

Now, for each U E q/, we have U <~W(U) and W(U) is a fuzzy open set. Then {W(U): U E q/} is a fuzzy open cover of X. By fuzzy almost compactness of X, there is a finite subcollection q/0 of q/ such that lx = U{cl W(U): U E q/0} = U{g-cl u: u E q/o}. Conversely, let the given condition hold. Then a given fuzzy open cover q/ of X is also a fuzzy cover of X by fuzzy a-open sets. By hypothesis, there is a finite subset q/0 of q/ such that lx = U{cc-cl U: u E ~//0}. By Lemma 8 and Remark 9, lx = [_J{cl U: U E q/0}. Hence X is fuzzy almost compact.

Definition 11 (Mukherjee and Ghosh [8]). An fts (X, T) is said to be fuzzy almost regular if for each fuzzy regular open set V in X (i.e. V = i n t c l V [2]) with x~ q V, there exists a fuzzy regular open set U such that x~ q U ~

Proof. Let an fts X be fuzzy almost compact and q/=-{Ui: i E A} be a fuzzy cover of X by fuzzy O-open sets. For each x E X and each n E N ( = the set of natural numbers), there exists Ux~Eq/ such that xl/nqU~. Then xl/nqVxn<~clVx"<~U~, for some fuzzy open set Vxn. Now, Vxn(X) + 1In > 1 ~ V~(x) > 1- 1In ~ sup,EN Vxn = 1. Put V~= U,~j ~n, so that V~(x)= 1, for each x EX. Hence {V~": x E X and h E N } is a fuzzy open cover of X. By fuzzy almost compactness of X, there exists a finite subset {x~,x2.... ,xk} of X and finitely many positive integers n l,n2,...,nk such that lx = U~=l cl vxin' " Then U ik= 1 u xi~ = ix and this proves the theorem. []

Theorem 14. If in a fuzzy almost regular space X, every fuzzy cover of X by fuzzy O-open sets has a finite subcover, then the space X is fuzzy almost compact. Proof. Let ~//be a fuzzy open cover of a fuzzy almost regular space. Then by Lemma 12, U ' = {intcl U: U E q/} is a fuzzy cover of X by O-open sets. By the given condition, there exists a finite subset q/0 of q/ such that lx = U{int (cl u ) : u E q/o} ~ U{cl u: u E q / o } . Thus, lx = U{clU: u E q/o} and X becomes fuzzy almost compact. [] From the last two theorems we obtain:

Theorem 15. A fuzzy almost regular space X isfuzzy almost compact iff every fuzzy cover of X by fuzzy O-open sets has a finite subcover. Definition 16. A fuzzy point x~ in an fts (X, T) is called a Ce-accumulation point of a fuzzy set A in X if for each fuzzy O-open q-nbd U of x~, IsuppAI--I{Y EX: A(y) + U ( y ) > 1}1,

M.N. Mukherjee, R.P. Chakraborty / Fuzzy Sets and Systems 98 (1998) 207-210

210

where for a fuzzy set A in X, supp A denotes the support (i.e. {x EX: A(x) >0}) of A, and for a subset B of X, by [BI we mean the cardinality of B.

Hence, by (3) we get m

[suppA[~< UAte,,, <[suppA[, Ii=1

Theorem 17. A necessary condition for an fts X to

be fuzzy almost compact is that every fuzzy set A in X with [supp A [/> d, (= the cardinal number of the set of integers) has a Co-accumulation point. Proof. Let A be a fuzzy set in a fuzzy almost compact space X such that [supp A[ ~>d, and if possible, suppose A has no Co-accumulation point in X. Then for each x E X and each n E N (=the set of naturals), there is a O-open q-nbd Ux~ of the fuzzy point xj/n (with support x and value 1/n) such that I{x EX: A(x) + Uff(x)> 1}[ < Isupp AI.

(2)

Now, since U~(x)+ 1/n > 1, it follows that {Uff: x E X and n E N} is a fuzzy cover of X by fuzzy O-open sets. By Theorem 13 there exist a finite subset {xl,x2 ..... xm} of X and a finite subset {n l, n2 ..... nm} of N such that Uim__~Uxn/= IX. Now,

x E supp A ~ Ux,nk (X ) = 1,

for some k ( l <~k <~m )

nk

gx~ (x) + A(x) > 1 =>x E {y EX: A ( y ) + Ux,nk ( y ) > 1} = Ate,,~ (say). As m

Au., C_UAt1.~, i=1

we have m

(3)

suppA C_UAc~,. i=l

But [Au,il<[suppA I

by(2) for i = l , 2 , . . . , m .

Thus i=UlAU"' = l~

IsuppAI.

which is a contradiction. This proves the result.

[]

Remark 18. The converse of the above theorem is false. In fact, for a finite set X with fuzzy topology T = [0, 1ix, the condition of the theorem is vacuously satisfied; but the fuzzy open cover T - {Ix} of X has no finite proximate subcover proving that the fts (X, T) is not fuzzy almost compact.

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