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Mathematica Bohemica Samajh, Singh Thakur; Ratnesh Kumar Saraf α-compact fuzzy topological spaces Mathematica Bohemica, Vol. 120 (1995), No. 3, 299–303 Persistent URL: http://dml.cz/dmlcz/126002 Terms of use: © Institute of Mathematics AS CR, 1995 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz 120(1995) MATHEMATICA BOHEMICA No. 3, 299-303 a-COMPACT FUZZY TOPOLOGICAL SPACES S.S. THAKUR, R.K. SARAF, Jabalpur (Received March 23, 1994) Summary. The purpose of this paper is to introduce and discuss the concept of acompactness for fuzzy topological spaces. Keywords: fuzzy topological spaces, compactness, a-compactness, a-open, fuzzy acontinuity AMS classification: 54A40 1. INTRODUCTION The concept of fuzzy sets and fuzzy set operations was first introduced by Zadeh in his classical paper [7]. Subsequently several authors have applied various basic concepts from general topology to fuzzy sets and developed a theory of fuzzy topological spaces. The concept of a-compactness for topological spaces has been discussed in [5]. The purpose of this paper is to introduce and study a-compactness for fuzzy topological spaces, thus filling a gap in the existing literature on fuzzy topological spaces. 2. DEFINITIONS Throughout this paper (X, T) will denote a fuzzy topological space. If A is a fuzzy set in a fuzzy topological space then the closure and interior of A will be as usual denoted by C\A and Int A, respectively. We now introduce the following definitions. D e f i n i t i o n 2 . 1 . Let A be a fuzzy subset of a fuzzy topological space X. A is said to be fuzzy a-open if A C Int CI Int A. The set of all fuzzy a-open subsets of X will be denoted by Fa{X). 299 Definition 2.2. In a fuzzy topological space (X, T), a family v of fuzzy subsets of X is called an a-covering of X iff v covers X and v C Fa(X). Definition 2 . 3 . A fuzzy topological space (X, T) is said to be a-compact if every a-open cover of X has a finite subcover. D e f i n i t i o n 2.4. Let (X,T) and (Y, S) be fuzzy topological spaces. A mapping / : X -4 Y is called fuzzy a-continuous if the inverse image of each fuzzy open set in Y is fuzzy a-open in X. Definition 2 . 5 . A mapping / : X -> Y is said to be fuzzy a-irresolute if the inverse image of every fuzzy a-open set in Y is fuzzy a-open in X. D e f i n i t i o n 2.6. Let (X, T) and (Y, S) be fuzzy topological spaces and let T { be a fuzzy topology on X which has Fa(X) as a subbase. A mapping / : X -4 Y is called fuzzy ^-continuous if / : (X,T() -4 (Y,S) is fuzzy continuous; / is said to be fuzzy ^'-continuous if / : (X,T{) -4 (Y,S{) is fuzzy continuous. 3. RESULTS T h e o r e m 3 . 1 . Let (X, T) and (Y, S) be fuzzy topological spaces and let T ? be a fuzzy topology on X which has Fa(X) as a subbase. If f: (X,T) -¥ (Y,S) is fuzzy a-continuous, then f is fuzzy ^-continuous. P r o o f . Let / be fuzzy a-continuous and let V 6 S. Then / - 1 ( V ) e Fa(X) and so / - 1 ( V ) £ 7$. Thus / is fuzzy ^-continuous and this completes the proof. D T h e o r e m 3.2. Let (X,T) and (Y,S) be fuzzy topological spaces. Let T$ and 5 { be respectively the fuzzy topologies on X and Y which have Fa(X) and Fa(Y) as subbases. If f: (X, T) -4 (Y, S) is fuzzy a-irresolute then f is fuzzy £'-continuous. Proof. Let / be fuzzy a-irresolute and let V e S { . Then V = ( J C p | S;„. \ i 4=i ' and where S,„. 6 Fa(Y, S), l rЧv)^r ({j(n^)-ö(Qr^))Since / is fuzzy a-irresolute, / - 1 ( S i „ ; ) 6 Fa(X,T). T { and thus / is fuzzy ^'-continuous. 300 This implies that / X (K) e E T h e o r e m 3 . 3 . A fuzzy topological space X is a-compact if and only if every family of fuzzy a-closed subsets of X with finite intersection property has non-empty intersection. Proof. Evident. D T h e o r e m 3.4. Let (X,T) be a fuzzy topological space and T( a fuzzy topology on X which has Fa(X) as a subbase. Then (X,T) is a-compact if and only if (X ,T() is compact. P r o o f . Let (X,T() is a-compact. be compact. Then, since Fa(X) C T ? , it follows that (X, T) D T h e o r e m 3.5. Let (X, T) be a fuzzy topological space which is a-compact. each T^-closed fuzzy set in X is a-compact. Then P r o o f . Let U be any T r closed fuzzy set in X. Let {Vpi: ft e / } be a T r o p e n cover of U. Since X - U is T r o p e n , {V0i: ft e / } U (X - U) is a T r o p e n cover of X. Since X is T r c o m p a c t , by Theorem 3.4 there exists a finite subset Io C I such that X = \J{VPi : d e I0} U(XU). This implies that U C \J{V0I: Pi e Io}. Hence U is a-compact relative to X and this completes the proof. D T h e o r e m 3.6. Let a fuzzy topological space (X,T) be a-compact. Then every family ofT(-closed fuzzy subsets ofX with finite intersection property has non-empty intersection. P r o o f . Let X be a-compact. Let U = {B0i : ft e / } be any family of T r closed fuzzy subsets of X with finite intersection property. Suppose fli-Bft : ft e / } = 0. Then {X - B0i : ft e / } is a T r o p e n cover of X. Hence it must contain a finite subcover {X - BSij: j = l , 2 , . . . , n } for X. This implies that f]{B0ij: j = 1,2,... ,n} = 0 and contradicts the assumption that U has finite intersection property. D T h e o r e m 3.7. Let X and Y be fuzzy topological spaces and let f: X -¥ Y be fuzzy £' -continuous. If a fuzzy subset G of X is a-compact relative to X, then f(G) is a-compact relative to Y. P r o o f . Let {Vp.: ft e / } be a cover of f(G) by 5 r o p e n fuzzy sets in Y. Then {/" (Vp^: Pi e / } is a cover of G by T r o p e n fuzzy sets in X. G is a-compact 301 relative to X. Hence by Theorem 3.4, G is T ? -compact. subset I0 C I such that Gc(J{r1(VA):/9,e/o} So there exists a finite and so / ( G ) C {V0i: A 6 / „ } . Hence / ( G ) is T^-compact relative to Y. Thus / ( G ) is a-compact relative to Y and this completes the proof. D Corollary 3.8. Iff: (X,T) -» (Y, S) is a fuzzy £'-continuous and X is a-compact, then Y is a-compact. surjective function Corollary 3.9. If f: (X,T) -> (Y, S) is a fuzzy a-irresolute and X is a-compact then Y is a-compact. surjective function T h e o r e m 3.10. Let A and B be fuzzy subsets of a fuzzy topological space X such that A is a-compact relative to X and B is T^-closed in X. Then An B is a-compact relative to X. P r o o f . Let {Vp{: ft e 1} be a cover of A n B by Tj-open fuzzy subsets of X. Since X - B is a T^-open fuzzy set, {VPi :/3ieI}U(X-B) is a cover of A. A is a-compact and thus T^-compact relative to X. exists a finite subset I0 C I such that A C {}{VPi: fc e I0} U(X- Hence there B). Therefore AnB Cljffis.: A €Io}. Hence A n B is T^-compact. Therefore A n B is a-compact and this completes the proof. D References [1] C.L. Chang: Fuzzy topological spaces. J. Math. Anal. 24 (1968), 182-190. [2] O. Njdsted: On some classes of nearly open sets. Pacific J. Math. 15 (1976), 961-970. [3] S. N. Maheshwari and S. S. Thakur: On a-irresolute mapping. Tamkang J. Math. 11 (1980), 209-214. [4] S. N. Maheshwari and S. S. Thakur: On a-continuous mapping. J. Ind. Acad. Math. 7 (1985), 46-50. [5] S. N. Maheshwari and S. S. Thakur: On a-compact spaces. Bull. Inst. Math. Acad. Sinica 13 (1985), 341-347. [6] A.S. Mashhour, I. A. Hassanian, S.N. El-deeb: a-continuous and a-open mappings. Acta Math. Sci. Hungar. 41 (1983), 213-218. [7] L. A. Zadeh: Fuzzy sets. Inform and control 8 (1965), 338-353. Authors' addresses: S. S. Thakur, Department of Applied Mathematics, Government Engineering College, Jabalpur (M.P.), India 482011, R.K. Saraf Department of Mathematics, Government Autonomous Science College, Jabalpur (M.P.), India 482001. 303