On Maximal and Minimal Fuzzy Sets in I-Topological Spaces

In this paper, the unit interval 0, 1 will be denoted by I. LetX be a nonempty set. A member of I is called a fuzzy subset of X 1 . Throughout this paper, for A,B ∈ I we write A ≤ B if and only if A x ≤ B x for all x ∈ X. By A B, we mean that A ≤ B and B ≤ A, that is, A x B x for all x ∈ X. Also we write A < B if and only if A ≤ B and A/ B. If {Aj : j ∈ J} is a collection of fuzzy sets in X, then ∨ Aj x sup{Aj x : j ∈ J}, x ∈ X, and ∧ Aj x inf{Aj x : j ∈ J}, x ∈ X. If r ∈ 0, 1 , then rX denotes the fuzzy set given by rX x r for all x ∈ X. The complementA of a fuzzy setA inX is given byA x 1−A x , x ∈ X. IfU ⊆ X, thenXU denotes the characteristic function of U. In this paper, we will follow 2 for the definitions of I-topology, the product Itopology, the direct and the inverse images of a fuzzy set under maps and their notations, fuzzy continuity, and fuzzy openness. A fuzzy set p defined by