正则和正规空间的若干判定定理

ON SOME PREDICATIVE THEOREMS OF THE REGULAR AND NORMAL SPACE

本文给出了正则和正规空间的4个判定定理:定理1 拓扑空间 X 为正则空间,当且仅当对于 X 中的任一点x 以及 x 中不含 x 的任一闭集 B,x、B 分别有闭邻域 U、V,使得U∩V=.定理2 拓扑空间 X 为正规空间,当且仅当对于 X 中的任意不相交的闭集 A、B、A、B分别有不相交的闭邻域 U、V,使U∩V=.定理3 拓扑空间 X 为正则空间,当且仅当对 X 中的任一点 x 以及不含点 x 的任意闭集B,分别有 x,B 的闭邻域 U、V,使得 i(U)∩i(V)=.定理4 拓扑空间 X 为正规空间,当且仅当对 X 中的任意两个不相交的闭集 A、B,A、B 分别有闭邻域 U、V,使得 i(U)∩i(V)=.

In this paper we prove four theorems,all part of a program to pronounce regular and normal space. Theorem 1 A topological space X is regular if and only if each point x in X and any open neighborhood U of x,there is a closed neighborhood V of x such that VU. Theorem 2 A topological space X is normal if and only if each closed set A in X and any open neighborhood U of A,there is a closed neighborhood V of A such that V U. Theorem 3 A topological space X is regular if and only if any point x in X and any closed sct B in X,x∈B,there are disjoint closed neighborhoods of x and of B. Theorem 4 A topological space X is normal if and only if any disjoint closcd sets A and B in X,there are closed neighorhoods U,V of A and of B,such that,U∩V=