Proc. Pakistan Acad. Sci. 42 (2): 000-000.2005 . E-mail: ¹o_r_sayed@yahoo.com ²noiri@as.yatsushiro-nct.ac.jp FUZZY γ -CONVERGENCE AND FUZZY γ C -CONVERGENCE OF NETS AND FILTERS IN FUZZIFYING TOPOLOGY 1 O. R. Sayed and 2 T. Noiri 1 Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516 Egypt and 2 Department of Mathematics, Yatsushiro National College of Technology, Yatsushiro, Kumamoto, 866-8501 Japan Received June 2004, accepted December 2004 Communicated by Prof. Dr. M. M. Qurashi Abstract: In this paper, the theory of γ -convergence and γ c -convergence on nets and filters is established in fuzzifying topology. Some important and interesting results in fuzzifying topology are obtained by means of the theory. Keywords: Fuzzy logic; fuzzifying topology; convergence, fuzzifying γ -open sets and fuzzifying γ c -open sets. 2000 Mathematics Subject Classification: 54A40 Introduction In [4-6], Ying elementally establ- ished the notion of fuzzifying topology with the semantic method of continuous valued logic. He discussed the neighbor- rhood structure of a point and the conv- ergence of nets and filters in this new fra- mework. Also, he presented the concepts of interior, closure and continuity and th- eir fundamental properties in fuzzifying topology. In [2] the concept of fuzzy γ - open sets and fuzzy γ -continuity were introduced and studied in fuzzifying top- ology. In [3], the concepts of fuzzy γ c - open sets and fuzzy γ c -continuity in fu- zzifying topology were presented and by making use of these concepts, some dec- omposition of fuzzifying continuity were introduced. The main purpose in the present paper is to introduce the theory of γ -co- nvergence and γ c -convergence on nets and filters in fuzzifying topology. Furthermore, we provide some interestin- g characterizations concerning γ -contin- uity, γ c -continuity and γ -Hausdorff sp- aces by making use of the γ -convergen- ce and γ c -convergence theory of nets in fuzzifying topology. Preliminaries We present the fuzzy logical and corresponding set theoretical notations [4,5] since we need them in this paper. For any formula ϕ , the symbol ] [ ϕ means the truth value of ϕ , where the set of truth values is the unit interval [0,1]. We write ⊂ ϕ if 1 ] [ = ϕ for any interpretation. Also, ) ( X ℑ is the family of all fuzzy sets in X. The truth valuation rules for primary fuzzy logical formulae and corresponding set theoretical notations are: