Cook-fischer Diagonal Condition And Separability Of Fuzzifying Convergence Space

In this paper, a concept of fuzzifying Cook-Fischer diagonal condition isproposed under the framework of fuzzifying mathematics. After discussing therationality of fuzzifying Cook-Fischer diagonal condition, it is concluded thata fuzzifying convergence structure must be pretopological whenever it satisfesthe fuzzifying Cook-Fischer diagonal condition. At the same time, the defnitionof the separability of fuzzifying convergence space is given, and the characteri-zations of T0-fuzzifying convergence space, T1-fuzzifying convergence space andT2-fuzzifying convergence space is discussed. Further, the paper shows that afuzzifying convergence structure is topological if and only if it satisfes fuzzifyingCook-Fischer diagonal condition. Finally the author fnds that the regularityof fuzzifying convergence structures can be described by Cook-Fischer diagonalcondition, and the characterizations of fuzzifying regular convergence spaces isprovided.The passage is consisted of fve sections.The frst section is preface. The author introduces the background anddevelopment condition of fuzzifying Cook-Fischer diagonal condition, providesthe main discussing content.The second section is preliminaries. In this part, the author leads into somenecessary marks. What is more, in order to discuss fuzzifying Cook-Fischerdiagonal condition, some necessary concepts and formulas are reviewed.The third section researches about fuzzifying Cook-Fischer diagonal condi-tion. First of all, fuzzifying Cook-Fischer diagonal condition is presented, andits correction is testifed. And then, the author proposes the innovation of thispaper: the pre-topological of fuzzifying convergence space can be described byfuzzifying Cook-Fischer diagonal condition. Secondly, the author reviews the (γ) condition what is proposed by Cook and Fischer, and raises fuzzifying (γ)condition, in this paper, it is called (FYNC). And then, the author proposes another innovation of this paper: the fuzzifying (γ) condition can be describedby fuzzifying Cook-Fischer diagonal condition.The forth section discusses the separation of fuzzifying convergence space.In this part, T0-fuzzifying convergence space, T1-fuzzifying convergence space,T2-fuzzifying convergence space is put forward, their characters and relationshipsare discussed.The ffth section is discussing the regulation of fuzzifying convergence space.In this part, the author proposes regular fuzzifying convergence space, debatesits characters and testifes that it is equal to regular fuzzifying topological space.