Studies Of Separation Axioms In Fuzzy Topology

As one of the fundamental branches of mathematics,from its foundation up till now,more than 100 years has passed in the history of general topology.Though it does not have as the long history as some other divisions of mathematics,for instance, Analytics,Algebra,Euclidean geometry and Number theory,general topology has made fast paces,especially from the 1950's to the 1970's,and is now becoming quite mature.As modern mathematics began to enter the forbidden region in the past+fuzzy research,in 1968,L.A.Zadeh,an American mathematician,first proposed the fuzzy set theory.And then this theory with the subsequent fuzzy logic theory consists of a basis for the modern fuzzy mathematics.As fuzzy mathematics extends the classical set theory,kinds of mathematical structures based on the fuzzy set theory come into being.Through over 40 years' joint efforts of scholars,domestic and abroad,gratifying progresses have been made in fuzzy topology,fuzzy analysis and fuzzy algebra.Fuzzy topology mainly concentrates on the L topological space,fuzzifying topological space andâ… -fuzzy topological space.Since topological space is a special case of fuzzifying topological space,many classical theories of topological space can be extended in fuzzifying topological space.And as a special case ofâ… -fuzzy topological space,fuzzifying topological space has much more complicated properties than general topological space.This paper studies fuzzifying topological space andâ… -fuzzy topological space using continuous-valued logic system semantic method.It mainly discusses preseparation axioms of these two fuzzy topological spaces,which further enrich and develop the basic theories of fuzzy topological space.This paper is organized as follows:(1)In general topology,the pre-open sets and pre-closed sets are important definitions.We define the pre-kernel and pre-θneighborhood of fuzzifying topological space using continuous-valued logic semantics and the open set in fuzzifying topological space as a tool,and discusses some important properties of pre-θcontinuous mapping.In particular,we give the definition of pre-R₀ separation axiom in fuzzifying topological space by pre-open set,pre-θclosure,pre-closnre.We have proved that pre-R₀ separation axiom in fuzzifying topological space is weaker than pre-T₁ separation axiom,while it has a good property that it can be described by only pre-closure and pre-kernel of a single point set.(2)Inâ… -fuzzy topological space,by R-neighborhood system theory,we define pre-open set and pre-closed set ofâ… -fuzzy topological space and then pre-R-neighborhood system ofâ… -fuzzy topological space.In this paper we discuss the properties of pre-R-neighborhood system,and use pre-R-neighborhood system to give a further discussion on the pre-continuous mapping,pre-θcontinuous mapping, pre-net convergence,pre-separation axiom and pre-R₀ separation axiom.From these discussions,we can infer that in theâ… -fuzzy topological space with fuzzifying topological space as a special case,many properties of the fuzzifying topological space have effective augmentation.