Some Research And Applications Of The Topology Of Defined By Nonstandard Analysis

In this paper, the topology of the nonstandard definitions is given in nonstandard enlargement model, and based on it the variety of concepts and conclusions in topological space are described and characterized. From nonstandard analysis, the original conclusions in topological space are researched and developed. On the one hand, the new definition of topology is consistent with the general; on the other hand, the new definition of the method is a way to study topology, many concepts and conclusions in topological space are simplified, and the essence of their nature more clearly.In the first part, firstly, the nonstandard analysis theory is briefly introduced. Using the axioms in nonstandard analysis, the nonstandard analysis is axiomatic approached. Secondly, some properties of nonstandard model is discussed, such as transitivity, Boolean properties, etc. Finally, the nonstandard enlargement model is studied, and some sufficient and necessary conditions are given.In the second part, the topology of the nonstandard definitions is given, the new definition of topology is proved equivalent to the general, and basis on it, neighborhood, monad ,accumulation, closed sets, are defined, This will be a major tool in this paper.In the following of this paper, the nonstandard definitions of the axioms separation, compactness, Moore-Smith convergence theory in topological space are given, and some related theorems and conclusions are proved by these characterizations.By the research this paper, not only the concept and conclusions in topology can be more clear and simple, but also ream of nonstandard analysis can be extended.