∆-OPEN SETS AND δ-SEMI CLOSED SETS IN FUZZY TOPOLOGICAL SPACES

Abstract

The study of fuzzy topological spaces was introduced by Chang in 1968, following the discovery of fuzzy sets by Zadeh. So far, several types of open sets and closed sets in fuzzy topological spaces have been documented in the literature. Some of them are fuzzy pre-open sets, fuzzy α-open sets, fuzzy β-open sets, fuzzy regular open sets, and fuzzy semi-open sets. In this abstract, we describe two new types of sets, namely fuzzy ∆-open sets and fuzzy δ-semi-closed sets, and discuss the properties of these sets. For a non-empty set X, a fuzzy topology is a family τ of fuzzy subsets in X satisfying the following conditions: 0_X ,1_X∈ Τ ; the finite intersection of members of τ is a member of τ; and the arbitrary union of members of τ is a member of τ. We call the pair (X,Τ) a fuzzy topological space. Also, the elements of τ are called fuzzy open sets. First, we define fuzzy ∆-open set in fuzzy topological spaces. A subset D of a fuzzy topological space X is called fuzzy ∆-open if D=(A∧B^C)∨(B∧A^C), where A and B are fuzzy open sets. When the set D is called fuzzy semi ∆-open, the above equation holds with the fuzzy semi-open sets A and B. Similarly, we can define the following sets: fuzzy pre ∆-open / fuzzy α-∆-open / fuzzy β-∆-open. Next, we show that every fuzzy ∆-open set is fuzzy semi ∆-open. However, the converse need not be true in general. Next, we define fuzzy δ-open sets in fuzzy topological spaces. A subset A is called fuzzy δ-open if for every x∈A, there exists a regular open set G such that x∈G≤A. A subset A is called fuzzy δ-semi closed if there exists a fuzzy δ-closed set F such that int(F)≤A≤F. Next, we show the following: A subset A is called fuzzy δ-semi open if and only if X\A is fuzzy δ-semi closed; every fuzzy δ-closed set is fuzzy δ-semi closed; every fuzzy δ-semi closed set is fuzzy semi-closed; a fuzzy set A is fuzzy δ-semi closed if and only if int(δcl(A))≤A. Finally, we show that the intersection or union of two fuzzy δ-semi closed sets is also fuzzy δ-semi closed.