Some Generalizations Of Minimal Fuzzy Open Sets

Acta Math. Univ. Comenianae Vol. LXXV, 1(2006), pp. 107–117

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SOME GENERALIZATIONS OF MINIMAL FUZZY OPEN SETS
SAMER AL GHOUR Abstract. Two new minimality concepts are defined and investigated in fuzzy spaces. Relationships between them and the known one are obtained. Several results concerning them are obtained. The results especially deal with products, separation axioms, maps, and fuzzytopologically generated topologies.

1. Introduction In this paper, the symbol I will denote the unit interval [0,1]. Let X be a nonempty set. A fuzzy set in X [11] is a function with domain X and values in I; that is, an element of I X . Throughout this paper, for λ, µ ∈ I X we write λ ≤ µ iff λ (x) ≤ µ (x) for all x ∈ X. By λ = µ we mean that λ ≤ µ and µ ≤ λ, i.e., λ (x) = µ (x) for all x ∈ X.If {λj : j ∈ J} is a collection of fuzzy sets in X, then ( λj ) (x) = sup {λj (x) : j ∈ J}, x ∈ X; and ( λj ) (x) = inf {λj (x) : j ∈ J}, x ∈ X. If r ∈ [0, 1] then rX denotes the fuzzy set given by rX (x) = r for all x ∈ X; that is, rX denotes the “constant” fuzzy set of level r. In this paper we shall follow [10] for the definitions of fuzzy topology, first countable fuzzy spaces, the productfuzzy topology, the direct and the inverse images of a fuzzy set under maps, fuzzy continuity and fuzzy openness. A fuzzy set p defined by p (x) = t if x = xp 0 if x = xp

where 0 < t < 1 is called a fuzzy point in X, xp ∈ X is called the support of p and p (xp ) = t the value (level) of p [10]. Two fuzzy points p and q in X are said to be distinct iff their supports are distinct, i.e., xp = xq . Afuzzy set p defined by p (x) = 1 if x = xp 0 if x = xp

is called a fuzzy crisp point in X [10]. The set of all fuzzy points in a set X will be denoted by F P (X). A fuzzy set that is either a fuzzy point or a fuzzy crisp point is called fuzzy singleton [6].
Received January 8, 2005. 2000 Mathematics Subject Classification. Primary 54A40. Key words and phrases. Fuzzy topological spaces, minimality,product fuzzy topology, fuzzy separation axioms, and fuzzy maps.

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In this paper, we follow [9] for the definition of ‘belonging to’. Namely: A fuzzy point p in X is said to belong to a fuzzy set λ in X (notation: p ∈ λ) iff p (xp ) < λ (xp ). If (X, ) is a fuzzy space and a ∈ [0, 1), then the set λ−1 (a, 1] : λ ∈ } is a topology on X (see [8]). This topology is denotedby a , and is said fuzzy topologically generated. A bijective function h : (X, 1 ) → (Y, 2 ); where (X, 1 ) and (Y, 2 ) are both fuzzy spaces; will be called fuzzy homeomorphism [9] iff h and h−1 are both fuzzy continuous. A non-empty open set A of a space (X, τ ) is called a minimal open set in X [7] if any open set in X which is contained in A is ∅ or A. The authors in [7] gave various resultsconcerning minimal open sets. In [2], the author studied minimal preopen sets and characterized them as singleton preopen sets. In [3] the authors extended the concept of minimal open sets to include fuzzy spaces. A fuzzy open set λ of a fuzzy space (X, ) is called minimal fuzzy open set [3] if λ is non-zero and there is no non-zero proper fuzzy open subset of λ, many interesting results concerningthis concept are obtained. In this paper, as a generalization of minimal fuzzy open sets we fuzzify minimality at some point by two methods, local minimality at a point in X and local minimality at a fuzzy point in X. Throughout this paper, for any set X, |X| will denote the cardinality of X. If λ is a fuzzy set in X, then the support of λ is denoted by S (λ) and defined by S (λ) = λ−1 (0, 1]. Theset of all minimal fuzzy open subsets of a fuzzy space (X, ) will be denoted by min (X, ). 2. local minimal fuzzy open sets We start by the following two main definitions. Definition 2.1. Let (X, ) be a fuzzy space, x ∈ X, and λ ∈ x ∈ S (λ). Then λ is called a local minimal fuzzy open set at x if proper fuzzy open subset of λ containing x in its support. The set minimal fuzzy open sets at a point x...