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International Journal of Applied Information Systems (IJAIS) – ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 11 – No. 6, November 2016 – www.ijais.org Separation of Fuzzy Topological Space Mohammad Arshaduzzaman, PhD M.Sc (Mathematics) Associate Professor Department of Mathematics Al Baha University, Al Baha Kingdom of Saudi Arabia (KSA) ABSTRACT 2. THEOREM (I) In this paper I discussed about some separation properties of fuzzy topological space. The definition of fuzzy compactness does not hold good on the definition of fuzzy Hausdorff space, we introduce a new definition of proper compactness and prove some interesting results related with it. Every fuzzy T2- space is a fuzzy T1- space. Keywords Fuzzy set, Fuzzy topological space, Fuzzy Hausdorff space, compactness, T2- space. Let X be any set. A fuzzy set A in X is characterized by A : X 0,1 A fuzzy singleton p in X is p a fuzzy set with the membership function p (X) y if x x 0 defined by = 0 otherwise where y (0,1). x 0 Hence pAiff p x 0 A x 0 (i) fuzzy set in X, if (ii) with p Gpqn qn Gpqn for all n, as (X, T) is fuzzy T 2. If P Gpqn then p is a p x q 0 and p x p 1 closed set with p Vpqn qP' p' Consequently, p' is 1 then there exist open sets and Gp and pGp and qGq,pGp p x p q x q there exist an open Gp with nN p is closed. So, (X, T) is fuzzy T -space. an open set i.e. xp x q xp = xq and pqn can find a sequence of open sets So, P' is an open set with Gq, G Case (ii) : Let p be a crisp point and converges to zero, then we p x 0 A x 0 p, q X , If (p) x p Gq x p and Definition 1 : A fuzzy topological space is said to be Housdorff of fuzzy T2 if the following conditions are satisfied : For any qp' Case (i) : Let p be a fuzzy point in X, and let be arbitrary. Then, there exist an open set Gq containing q such that Hence Gq {p}. So, {p} being the union of open sets, is an open set. This implies that {p} is closed. 1. INTRODUCTION membership function Proof : Let (X, T) be a fuzzy T2- space. pGp but qGp We now introduce the concept of properly compact space in a fuzzy topological space and prove an analogous result of general topology which holds for a compact Hausdroff space.By Palaniappan[3] & Zadeh[4] Definition 4 : . By Srivastava & Lal [8]. Definition 2 : A fuzzy set A is said to be open for each , there exists an open pG A set G with . Out definition of open set is equivalent to that given by Change [1]. Definition 3 : A fuzzy topological space is said to be a fuzzy T1- space if the singletons are closed. By Hutton & Reilly [2]. G :iI 1 A family open sets in fuzzy topological space X is called a proper open cover of a fuzzy set A in X if for every x X there exists a member this family such that The family Gix of this family such that of Gix (x) A (x) Gi :iI is called a proper open sub cover of Gi :iI if it is itself a proper open cover of A. By Lowen[5]. 1 International Journal of Applied Information Systems (IJAIS) – ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 11 – No. 6, November 2016 – www.ijais.org Definition 5 : A fuzzy set A in a fuzzy topological space X is said to be properly compact if every proper open cover of A is reducible to a finite proper open sub cover. Then Fp is closed and We also have p x p Fp x p for all x q X p x p Fp x p (iv) By Lowen [6] & Rodabaugh[7]. Then, for all satisfying the condition (i), the family 3. THEOREM 2 F 'p Every properly compact set in a fuzzy T2- space is closed. Proof : Let A be a properly compact set in a fuzzy T2-space X. We choose a point with X is A x p Gp x p Thus, G for each pq : x q X exists G family with (iv) pq an open Gp with there exists a family of open sets for all of open sets x q X Then, p with establishes that A is closed. 4. CONCLUSION (iii) X such : x X From the above theorems and definitions it is clear that A fuzzy set in a fuzzy topological space is said to be properly compact if every proper open cover of a fuzzy set is reducible to a finite proper open sub cover and every properly compact set in a fuzzy T2-space is closed. Thus the definition of fuzzy compactness does not hold good on the definition of fuzzy Hausdorff space. 5. REFERENCES there exists a A x q Gpq x q sup A x q x q Gpq x p [1] Chang, C.L. : Fuzzy topological spaces, J.Math. Anal. Appl. 24, 182-190, 1960. [2] Hutton, B & Reilly, I. : Seperation axioms in fuzzy topological space, Fuzzy Sets & systems 3, 99-104, 1980. [3] Palaniappan, N. : Fuzzy topology, Narosa Publ. House, New Delhi, 2002. [4] Zadeh, L.A. : Fuzzy sets, Inform, Control, 8, 338-353. [5] Lowen R : Fuzzy topological spaces and fuzzy compactness,J. Math. Anal. Appl.,56 (1976), pp.621–633. A Gpq This implies that X, satisfies the conditions (v) & (vi). we have A Fp (ii) A x q Gpq x q with So, there p x p Gp x p and p A x q Inf Fp x q for all x q X. (i) T2, F where p p x p A x p since is open in Consequently (v) x q X G Thus, the family [6] Lowen R: Compact Hausdorff fuzzy topological spaces are topological, Topology and Appl., 12 (1981), pp. 65– 74. G [7] Rodabaugh S :The Hausdorff separation axiom for fuzzy topological spaces,Topol. Appl.,11 (1980),pp. 319–334. pq : x q X is a proper cover of A. Since A is properly compact, there exists a finite subfamily, say, ,Gpq2 ,....., Gpqmof Gpq : xqXofGpq : x q X pq1 m A Gpqk .Then A Gpqk Fp (sup pose) K 1 with [8] Srivastava R, Lal S N, and Srivastava A K, “Fuzzy Hausdorff topological spaces,” Journal of Mathematical Analysis and Applications, vol. 81, no. 2, pp. 497–506, 1981. 2