具有特殊结构的可度量化空间的性质研究

Properties Study on the Properties of Metrizable Spaces with Special Structure

摘要: 可度量化空间是分析学、一般拓扑学等研究的重要对象，在众多可施加在拓扑空间上的分离公理中，Hausdorff空间蕴涵了序列、网和滤子的极限的唯一性，是最常使用和讨论的。本文通过运用两个基本的度量化定理得到度量化的Hausdorff空间，在此基础上研究构造几种常见的可分空间的拓扑结构，通过论证得到，经构造的可分度量空间满足群、环、PID的相关条件。

Abstract: The metrizable spaces are important objects of analysis, general topology and other studies. Among many separation axioms that can be imposed on topological spaces, Hausdorff spaces contain the uniqueness of the limits of sequences, nets and filters, which are most often used and discussed. In this paper, two basic metrization theorems are used to obtain the metrized Hausdorff space. On this basis, several common topological structures of separable spaces are studied and constructed. Through demonstration, the constructed separable metric space meets the relevant conditions of group, ring and PID.