关于第一可数拓扑空间的一个注解

A note on the nets in first-countable spaces

本文讨论了第一可数拓扑空间中收敛网的特征.引入了全有序网和严格收敛网的概念并利用它们给出了第一可数空间的一个充要条件:设(X,T)是T1空间则下列结论是等价的:1)(X,T)是第一可数拓扑空间,2)对于任何一点x0∈X,(X,T)中有一个全有序的x0点邻域基,并且每一个严格收敛于x0的全有序网必有一个收敛到x0点的子序列.

Some characteristic of convergent nets in first-countable spaces are discussed. Totally ordered-net and the strictly convergence of nets is introduced and the following necessary and sufficient condition describing the character of first-countable with term of nets is given： Let（X,T） be a topological space. If （X,T） is a T1 space, the following statements are equivalent： 1 ） （ X, T） is a first-countable space. 2 ） （ X, T） has a totally ordered-base of neighborhoods at each x0 ∈ X and every totally ordered-net { x0 ：δ∈ D/which strictly converges to x0 must have a subsequence{ yk：k∈ω} satisfying Yk→x0 when k→∞.