函数空间的拓扑分类

项目名称： 函数空间的拓扑分类

项目编号： No.11471202

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 杨忠强

作者单位： 汕头大学

项目金额： 75万元

中文摘要： 无限维拓扑学是拓扑学的一个重要分支，为拓扑学提供了很多新方法和新问题。它与泛函分析，集值分析，微分方程等数学学科有着密切的联系。本项目希望在已完成的三项国家自然科学基金的基础上，研究函数空间的拓扑分类问题。为此，需要建立特定的广义度量空间和度量空间之间的关系，给出描述集合论中我们所需要的一些性质，探讨构造连续函数和同胚的新方法, 构造各种拓扑空间类的吸收系统。通过这些工作，我们希望解决：在Fell拓扑下可度量化的部分函数空间的拓扑分类问题，在Hausdorff度量拓扑下函数空间的拓扑分类问题，在Lp度量下的模糊数空间的拓扑分类问题，给出这些结果在集值分析，模糊数学等学科中的应用，为一般拓扑学提出新的有意义的问题。这些问题的深入系统的研究和解决，不仅大大促进了无限维拓扑学的发展，同时也为无限维拓扑学在其他相关学科中相关问题的研究提供新的理论框架和有效工具，有重要的理论意义和潜在的应用价值。

中文关键词： 拓扑分类；函数空间；超空间；无限维拓扑学；广义度量空间

英文摘要： Infinite-dimensional topology is one of the most important branches of topology, which supplies lots of new methods and problems for topology. It is closely connected to others mathematical disciplines,such as functional analysis,set-value analysis and differential equations. The project will discuss the topological classification problem on function spaces on the basis of three finished projects which are under the support of the National Natural Science Fund of China. In order to complete these works, we need to establish the relation between specific general metric space and metric space, and give some properties of set theory we need, also we provide some new methods to construct continuous functions and homeomorphisms and construct absorbers of various classes of topological spaces. We want to know the partial topological classification problem of metrizable function space under the Fell topology, the topological classification problem of the function space under the Hausdorff metric topology and the topological structure of fuzzy number space under Lp metric. We hope in this project may give some applications of our results in set-valued analysis and fuzzy mathematics and new problems for general topology. These works will not only make a contribution in infinite- dimensional topology but also may supply new knowledges and tools for infinite-dimensional topology applying to other disciplines. We think that the project has its important values in theory and applications.

英文关键词： Topological classification;Function spaces;Hyperspaces;Infinite-dimensional Topology;Generalized metric spaces