Some Applications Of Nonstandard Analysis In Fuzzy Topology

In this paper, the monads of neighborhood structure in fuzzy topological space are defined in nonstandard enlargement model, and many concepts and conclusions in fuzzy topological space are described and characterized by nonstandard analysis. From nonstandard analysis, the original conclusions in fuzzy topology are researched and developed. On the one hand, the intrinsic characterizations of fuzzy topology can be discovered by nonstandard analysis, and the closed, initial relationship between fuzzy topology and general topology can be found, on the other hand, the level structure of fuzzy topology can be shown — through that the concepts in fuzzy topological space are characterized by nonstandard analysis - it is different from general topology.In the first part, firstly, the nonstandard analysis theory is briefly introduced. Using the axioms in nonstandard analysis, the nonstandard analysis is axiomatic approached, and the existence of nonstandard model and the consistence of axioms in nonstandard analysis are proved by the construction of nonstandard model. Secondly, some properties of nonstandard model are discussed, such as transitivity, Boolean properties, etc. Thirdly, the nonstandard enlargement model is studied, and some sufficient and necessary conditions are given.In the second part, first, the fuzzy set and its calculations are extended by nonstandard analysis - it can take the nonstandard analysis theory into fuzzy mathematics. Next, the nonstandard enlargement model is also brought into fuzzy mathematics by the concurrent principle. It makes nonstandard enlargement model withthe fuzzy form. At last, the definition of fuzzy topological space is introduced, and based on it, three neighborhood structures in fuzzy topology are discussed - neighborhood, quasi-neighborhood, remote-neighborhood. The N-monad, Q-monad and R-monad are defined respectively by the theory of monad in nonstandard analysis, and the approach theorems and relations among them are proved. It is the most important tool in this paper.In the following chapters of this paper, the Moore-Smith convergence theory, the axioms of separation and the compactness in fuzzy topological space are characterized respectively, and some related theorems and conclusions are proved by these characterizations.By the research in this paper, not only the concepts and conclusions in fuzzy topology can be more clear and simple, but also realm of nonstandard analysis can be extended.