Some Applications Of Nonstandard Analysis In Linear Topological Space And Set-valued Mapping

In the analysis field, topology is an important branch of mathematics. In this paper, linear topological ac a specific topological space in topological are mainly studied, our attention in this paper are focused on such properties of linear topological space. If space E is a linear space, also a topological space, and the algebraic operations in the linear space is continuous according to the topology. As a kind of important space E is called the topological space. The linear topological space may be viewed as a generalization of linear distance space. In this discussioned research about linear topological space, we all suppose the linear topological space is Hausdorff linear topological space. In this paper, linear topological space is study by the nonstandard analysis methods, the conclusions are as follow.(1) In the topological space, the nonstandard characterizations of denseness and nowhere denseness of set in topological space are obtained;(2) In this paper, the nonstandard characterizations of continuity, upper semicontinuity and lower semicontinuity of mapping are shown. The continuity of mapping in linear topological space are proved by the nonstandard analysis methods;(3) The nonstandard characterization of linear topological space are obtained and proved. In the linear topological space, some properties of convex set and convex hull are studied by the nonstandard analysis methods;(4) In the Hausdorff topological space, the nonstandard characterization of Vietoris topology in a family of set are shown. Next, two new definitions of C-monad and I-monad are obtained. At last, some properties of convergence of set-valued nets in Hausdorff topological space are studied by the nonstandard analysis methods;(5) By the nonstandard analysis methods, the upper semicontinuity, lower semicontinuity, continuity, weak lower semicontinuity in Hausdorff topological space are characterized. The nonstandard characterization of graph of set-valued mapping are shown. At last, some properties of continuity of set-valued mapping are studied by the nonstandard analysis methods.