Some Applications Of Nonstandard Analysis In Uniform Topological Space

In the mathematical field of topology, uniform space refers to a set of uniform structure, it is a special kind of topological space, it can be used to define a lot of additional structure of uniform property. There are close relationship between uniform space and topological space, metric space, therefore uniform space has become an important link connection topological space and metric space. In this article, uniform space is study by the methods of nonstandard analysis, the conclusions are as follows.(1) By using the definition of gage, some nonstandard characterizations, such as the uniform convergence of function in uniform space. It shows that if fn=> f on (X,u) where each fn is continuous on(X,u), f is continuous on(X,u).(2) TheU-microcontinuity, U-equicontinuity, rs-continuity, and U-*-continuity of function are defined by methods of nonstandard analysis in uniform space. The equivalent propositions of describing U-microcontinuity by gage are proposed. It shows that f is uniformly continuous on (X,u) if and only if f is U-microcontinuous on (*X,*u),{fn|n∈N+} is U-equicontinuous on (X,u) if and only if fn is U-microcontinuous on (*X,*u) for each n∈*N+and if f is rs-continuous, then f is U-microcontinuous and so on, are shown. And the relationships among them are discussed.(3) Through the nonstandard characterizations of compact uniform space, it shows that if f is continuous on the compact (X,u), f is uniformly continuous. fn=> f on the compact (X,u) if and only if{fnln∈N+} is U-equicontinuous. And the approximation theorem of function in uniform space are proved by the concept of U-microcontinuity and internal function theorem. The properties of compact mapping in uniform space are discussed further; which lays the foundations for researching uniform space the later.(4) It shows that the relationship between Cauchy filters and the monad of a uniformity, the cluster point of Cauchy net is researched through nonstandard characterizations. Cauchy net convergent is defined by using methods of nonstandard analysis, it showed that Cauchy net convergent if and only if a cluster point exists in Cauchy net, and further (X,u) is complete if and only if each Cauchy net, there is infinity p∈*D such that Sp is near-standard. Which generalizes the nonstandard characterizations of uniform space completeness.