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A HIT-AND-MISS HYPERSPACE TOPOLOGY ON THE SPACE OF FUZZY SETS Anil Kumar Department of Applied Sciences Lingayas Institute of Management and Technology Faridabad INDIA E-mail: [email protected] J.V.Ramani Department of mathematics Agra college, Agra INDIA E-mail: [email protected] Abstract A hit-and-miss type Vietoris topology is defined on the space of non-empty closed fuzzy sets of a fuzzy topological space and it is proved that if this space is compact then the fuzzy topological space is also compact. 2000 Mathematics Subject Classification: Primary 54A40. Keywords and phrases: fuzzy topological space, hyperspace 1 Introduction The concept of a fuzzy set was introduced by L.A.Zadeh in [12] and later the concept of a fuzzy topology by C.L.Chang in [5]. After this introduction numerous authors have investigated how the properties of topological spaces can be extended to fuzzy topological spaces like compactness, connectedness and separation, etc., which can be found in [4, 7, 8, 9, 11]. On the other hand, topologies on the spaces of non-empty closed subsets of a metric space(or a Banach space or a topological space) have been investigated by G.Beer and his colloborators with regard to the problems in convex analysis and optimization [1, 2, 3, 6]. Of these topologies(called the hyperspace topologies) Vietoris topology on the space of non-empty closed subsets of a metric space occupies a central place among all the other hyperspace topologies. Vietoris topology is also called as a hit-and-miss type of topology as it is generated by the sets which hit a finite number of open sets and misses a closed set. More precisely, if (X, d) is a metric space and CL(X) denotes the space of non-empty closed subsets of X then the Vietoris topology τV on CL(X) has basic open sets of the form [U ; V1 , V2 , . . . Vn ] = {A ∈ CL(X) | A ⊂ U, A ∩ Vi 6= φ, i = 1, 2, . . . , n} where n ∈ N and U, V1 , V2 , . . . , Vn are open sets in (X, d). In [10] L. Vietoris proved that the hyperspace 2X of all non-empty closed subsets of a topological space X equipped with the Vietoris topology is compact iff X is compact. He also proved in the same paper that the connectedness of 2X in Vietoris topology is equivalent to that of X. In this paper we define a topology of hit-and-miss type on the space of non-empty closed fuzzy sets of a T1 -fuzzy toplogical space and prove that if this hyperspace topology is compact then the fuzzy topological space must be compact. 2 Notations In this note X will always denote a non-empty (ordinary) set. Any function f from X to I = [0, 1] is called a fuzzy set in X or a fuzzy subset of X. For any x ∈ X, A(x) is called the grade of membership of x in X. The set 2 {x ∈ X | A(x) > 0} is called the support of A and denoted by SuppA. If a fuzzy set A takes only the values 0 and 1 then the set is called a crisp set. The fuzzy set A(x) ≡ 1 will be denoted by X. Sometimes we will also denote this by 1 or I. The fuzzy set A(x) ≡ 0 will be denoted by 0 and sometimes by φ. Let A = {Ai | i ∈ Γ} be a family of fuzzy sets. Then we have (∪Ai )(x) = W V ( i Ai )(x) = sup {Ai (x)} for each x ∈ X, (∩Ai )(x) = ( i Ai )(x) = Inf {Ai (x)} for each x ∈ X. We recall the definition of a fuzzy topological space from [5]. A fuzzy topology is a family T of fuzzy sets in X satisfying the following conditions: φ, X ∈ T , for A, B ∈ T we have A ∩ B ∈ T , and ∪Ai ∈ T for Ai ∈ T for all i ∈ Λ. In this case we say (X, T ) is a fuzzy topological space and T is called a fuzzy topology on X. Every member of T is called a fuzzy open set. A fuzzy set in X is called a fuzzy point iff it takes the value 0 for all x ∈ X except one, say y ∈ X. If its value at y is λ (0 < λ ≤ 1), then we denote this fuzzy point by Pyλ . Two fuzzy sets A and B in X are said to be intersecting iff there exist a x ∈ X such that (A ∩ B)(x) 6= 0. In this case we also write A ∩ B 6= φ. (X, T ) is called a fuzzy T1 -space iff every fuzzy point is a closed set([7]). A family A of fuzzy sets is a cover of a fuzzy set C iff C ⊂ ∪ {A | A ∈ A}. It is an open cover of C iff each member of A is an open fuzzy set. A fuzzy topological space (X, T ) is said to be compact iff each open cover of X has a finite subcover([5]). This compactness is also called quasi fuzzy compactness. We now make the following definition required for this paper. Definition 2.1 Let (X, T ) be a fuzzy topological space and let C(X) denote the collection of all the non-empty closed fuzzy sets in X. That is φ ∈ / C(X). Consider the topology on C(X) generated by the sets of the form [U ; V1 , V2 , . . . Vn ] = {A ∈ C(X) | A ⊂ U, A ∩ Vi 6= φ, i = 1, 2, . . . , n} where n ∈ N and U, V1 , V2 , . . . , Vn are open sets in (X, T ). We denote this topology on C(X) by V. 3 3 Main Theorem We start with the following Proposition 3.1 If (X, T ) has a finite open covering then C(X) also has a finite open covering in V. Proof. Let X ≤ m k=1 Tk , Tk ∈ T , k = 1, . . . , m be a open covering of X. That is 1 = I(x) ≤ max {T1 (x), . . . , Tm (x)} for each x ∈ X. That is max {T1 (x), . . . , Tm (x)} = 1 for each x ∈ X. Let A ∈ C(X). Then A 6= 0. That is, there exists a y ∈ X such that A(y) = λ 6= 0. Now max {T1 (x), . . . , Tm (x)} = 1 for each x ∈ X. Therefore for this y ∈ X there exists at least one j, 1 ≤ j ≤ m with Tj (y) = 1. This gives that (A ∩ Tj )(y) = min(A(y), Tj (y)) = min(λ, 1) 6= 0. That is A ∩ Tj 6= φ. Also A(x) ≤ I(x) for all x ∈ X. This means that A ∈ [I; Tj ] for some j. Hence S C(X) ⊂ m i=1 [I; Ti ]. Now we come to the main theorem of the paper. W Theorem 3.2 Let (X, T ) be a T1 -fuzzy topological space. Then if (C(X), V) is compact then (X, T ) is also compact. Proof. Let X ≤ α∈Λ Tα , Tα ∈ T for all α ∈ Λ. That is Supα∈Λ {Tα (x)} = 1, for each x ∈ X. This gives Tα (x) ≤ I(x) = 1 for all x ∈ X, for all α ∈ Λ. Let A ∈ C(X). Then there exists a x ∈ X such that A(x) = λ 6= 0. If λ 6= 1 then we can find a α0 such that 1 − λ < Tα0 (x) ≤ 1. That is A ∧ Tα0 6= φ. That is A ∈ [I; Tα0 ]. If λ = 1 then SupTα (x) = 1 = A(x). That is Tα (x) ≤ A(x). Or A ∧ Tα1 6= φ for some α1 . That is A ∈ [I; Tα1 ]. S Thus C(X) ⊂ α∈Λ [I; Tα ] which is an open cover of C(X). Since C(X) is S compact, we have C(X) ⊂ ni=1 [I; Ti ]. As (X, T ) is a T1 -space, every fuzzy point is in C(X). Let x ∈ X and 0 < λ < 1. Consider the fuzzy point Pxλ . S Then Pxλ ∈ C(X) ⊂ ni=1 [I; Ti ]. That is Pxλ ∈ [I; Tj ] for some j. We get Pxλ ∧ Tj 6= φ. That is, min(λ, Tj (x)) 6= 0. If Tj (x) = 1 we are done. Suppose Tj (x) < 1. If λ = P (x) < Tj (x) then 1 ≤ Tj (x) as this is true for all λ → 1. If Tj (x) ≤ P (x) = λ then Tj (x) = 0 because 0 < λ, which gives Tj (x) = 0, W a contradiction. Hence we must have Tj (x) = 1. This gives ni=1 Ti (x) = 1. W Hence X ≤ ni=1 Ti . That is (X, T ) is fuzzy compact. W 4 References [1] G. Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc., 108(1990) 117-126. [2] G. Beer and R. Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc., 327(1991) 795-813. [3] G. Beer and R. Lucchetti, Weak topologies for the closed subsets of a metrizable space, Trans. Amer. Math. Soc., 335(1993) 805-822. [4] G. Balasubramanian and V. Chandrasekhar, Fuzzy α-connectedness and fuzzy α-disconnectedness in fuzzy topological spaces, Vesnik Matematicki., 56(2004) 47-56. [5] C. L. Chang, Fuzzy topological spaces, J. math. Anal. Appl., 24(1968) 182-169. [6] J.J. 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