IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 1 (May. - Jun. 2013), PP 63-70 www.iosrjournals.org www.iosrjournals.org 63 | Page On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topological Spaces R.Usha Parameswari 1 , K.Bageerathi 2 , 1,2 Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur-628215, India. Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy - t-set, -SO extremely disconnected space analyse the relations between them. MSC 2010: 54A40, 03E72. Key words: Fuzzy - open, fuzzy - closed, fuzzy - semi open, fuzzy - semi closed, fuzzy - semi interior and fuzzy - semi closure, - t-set and fuzzy topology. I. Introduction The concepts of fuzzy sets and fuzzy set operations were first introduced by L.A.Zadeh [6] in his paper. Let X be a non empty set and I be the unit interval [0,1]. A Fuzzy set in X is a mapping from X in to I. In 1968, Chang [3] introduced the concept of fuzzy topological space which is a natural generalization of topological spaces. Our notation and terminology follow that of Chang. Azad introduced the notions of fuzzy semi open and fuzzy semi closed sets. And T.Noiri and O.R.Sayed[5] introduced the notion of -open sets and -closed sets. Swidi Oon[4] studied some of its properties. Through this paper (X, ) (or simply X), denote fuzzy topological spaces. For a fuzzy set A in a fuzzy topological space X, cl(A), int(A), A C denote the closure, interior, complement of A respectively. By 0 x and l x we mean the constant fuzzy sets taking on the values 0 and 1, respectively. In this paper we introduce fuzzy -semi open sets and fuzzy -semi closed sets its properties are established in fuzzy topological spaces. The concepts that are needed in this paper are discussed in the second section. The concepts of fuzzy -semi open and fuzzy -semi closed sets in fuzzy topological spaces and studied their properties in the third and fourth section respectively. Using the fuzzy - semi open sets, we introduce the concept of fuzzy - SO extremely disconnected space. The section 5 and 6 are dealt with the concepts of fuzzy -semi interior and -semi closure operators. In the last section, we define fuzzy -t-sets and discuss the relations between this set and the sets defined previously. II. Preliminaries In this section, we give some basic notions and results that are used in the sequel. Definition 2.1: A fuzzy set A of a fuzzy topological space X is called: 1) fuzzy semi open (semi closed) [2] if there exists a fuzzy open (closed) set U of x such that U A cl U ( int U A U). 2) fuzzy strongly semi open (strongly semi closed) [4] if A int( cl( int A)) (A cl( int( cl A))). 3) fuzzy -open (fuzzy -closed) [5] if A ( int( cl A)) cl( int(A)) ( A (cl(int(A))) (int (cl(A)))). Definition 2.2[7]: If λ is a fuzzy set of X and μ is a fuzzy set of Y, then (λ μ)(x, y)= min { λ(x), μ(y)}, for each XY. Definition 2.3[2]: An fuzzy topological space (X, 1 ) is a product related to an fuzzy topological space (Y, 2 ) if for fuzzy sets A of X and B of Y whenever C c A and D c B implies C c 1 1 D c A B, where C 1 and D 2 , there exist C 1 1 and D 1 2 such that C 1 c A or D 1 c B and C 1 c 11 D 1 c = C c 11 D c . Lemma 2.4 [2]: Let X and Y be fuzzy topological spaces such that X is product related to Y. Then for fuzzy sets A of X and B of Y, 1) cl (A B) = cl (A) cl (B) 2) int (A B) = int (A) int (B) Lemma 2.5[1]: For fuzzy sets λ, μ, υ and ω in a set S, one has (λ μ) (υ ω) = (λ ω) (μ υ)