On Supra Fuzzy Topological Spaces

Abstract

The fundamental concept of a fuzzy set was introduced by L. A. Zadeh [111] in 1965 to provide a foundation for the development of many areas of knowledge. Consequently, this provides a natural frame work for generalizing many algebraic and topological concepts in various directions such as fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy supra topology, fuzzy infra topology, fuzzy bitopology etc. many other branches of mathematics have been developed all over the world during the last five decades. In 1968, Chang [19] introduced the concepts of a fuzzy topological space by using the fuzzy set. Wong [105], Lowen [60], Hutton [48], katsaras[52], Ali[3], Pu and Liu[72], etc., discussed various aspects of fuzzy topological spaces. Ying [74] introduced fuzzifying topology and developed this in a new direction with the semantic methods of continuous valued logic. In the frame work of fuzzifying topology, Sinha [93] introduced and studied To-, T1-, T2(Hausdorff)-, T3(regular)-, T4(normal)-, separation axioms. A.S. Mashhour et al. [ 64] introduced and studied the concepts of the family of fuzzifying semi open sets, fuzzifying neighborhood structure of a point and fuzzifying semi-closure of a fuzzy set. A.S. Mashhour et al. [64] also introduced and studied the R-0 and R1 separation axioms and studied their relations with the T1 and Ti-separation axioms respectively. Also, in fuzzifying topology they introduced and studied semi-To-, semi-Ro-, semi-Ti-, semi-R1-, semi-T2(semi Hausdorff)-, semi-T3(semi regular)-, semi-T4(semi normal)-, separation axioms. In 1983, A.S. Mashhour et al. [64] introduced supra topological spaces and studies s-continuous function and s • -continuous functions. In 1987, M.E. Abd EL-Monsef et al. [1] introduced the fuzzy supra topological spaces and studied fuzzy supra continuous functions and characterized a number of basic concepts..........