Fuzzy Topological Spaces, Fuzzy Filter Space And Fuzzy Semi-uniform Convergence Space A Few Questions

Fuzzy topology, including general topology, is a new theory. Following orders in fuzzy topologies, the same property in a fuzzy topology has being studied in many ways, in other words, we can investigate a property of fuzzy topology from different logical systems. This paper focuses on three aspects-convergence structures in a fuzzy topological space; completions in fuzzy filter spaces; properties of fuzzy semiuniform convergence spaces.Chapter1is a introduction. Some basic concepts and conclusions in lattice theory, fuzzy topology as well as categorical theory is presented.Chapter2mainly introduces two convergence structures,which are both categorically isomorphic to (L, M)-fuzzy topologies. Speaking more clearly, firstly remote neighborhood systems, closure operators, neighborhood systems and interior operators in (L, M)-fuzzy topological spaces are introduced. Next, with the help of closure operators, it is proved that the category of topological (L, M)-fuzzy (molecule net) convergence spaces is isomorphic to the category of (L, M)-fuzzy topological spaces. Following neighborhood systems, it is shown that the category of topological (L, M)-fuzzy (fuzzy filter) convergence spaces is isomorphic to the category of (L, M)-fuzzy topological spaces. Finally, the special case of topological (L, M)-fuzzy (fuzzy filter) convergence structures is studied, and two new forms are presented—diagonal condition and Gahler's neighborhood condition, which are both characterizations of L-topologies.Chapter3focuses on a kind of completion of stratified (L, M)-probabilistic fil-ter spaces. Firstly the relationship between stratified (L, M)-probabilistic filter spaces and stratified (L, M)-filter spaces is investigated, and the relationship between stratified (L, M)-probabilistic Cauchy spaces and stratified (L, M)-Cauchy spaces is also presented. Furthermore, as a example of categorical properties of these spaces, it is shown that the category of stratified (L, M)-probabilistic filter spaces is a strong topological universe. Following these work, a sufficient and necessary condition is given for stratified (L, M)-probabilistic filter spaces to have completions. As an application of which, a sufficient and necessary condition is easily obtained for stratified (L, M)-filter spaces to have com-pletions. Finally, in a much similar way, a sufficient and necessary condition is given for stratified (L, M)-probabilistic Cauchy spaces to have completions. As an application of which, a sufficient and necessary condition is easily obtained for stratified (L, M)-Cauchy spaces to have completions.Chapter4is devoted to stratified L-semiuniform convergence spaces. Several subcat-egories are defined and the relationships between these categories are given. Furthermore, the completions of probabilistic semiuniform convergence spaces are constructed and a suf-ficient and necessary condition is given for probabilistic semiuniform convergence spaces to have completions. Finally, the relationships among stratified L-semiuniform conver-gence spaces, stratified L-filter spaces, stratified L-Kent convergence spaces and stratified L-fuzzy topological spaces are presented.Finally, the main work of this thesis is summarized, and the future work is given.