拓扑空间子集的最小开集和最大闭集

Smallest Open Set and Largest Closed Set of a Subset of Topological Space

讨论对于拓扑空间子集A是否存在包含A的最小开集和是否存在包含于A的最大闭集问题,证明了拓扑空间X是一个T1空间之充分且必要条件是,对于X的每一个子集A,X中存在包含A的最小开集(存在包含于A的最大闭集)当且仅当A是X的开集(闭集),同时给出几个例子说明了定理的条件.

This paper discusses the smallest open set and largest closed set in a subset of topological space,and show that if X is a T space ,and A is a subset of X, then there exists the smallest set containing A in the space （ the largest closed set contained in A ） if and only if A is a open set（closed set）. Several examples are giving at the same time to explain the conditions of the theorems.