点集拓扑中的连通性注记

Remarks on Connectedness in Point-set Topology

现行的点集拓扑教材中关于局部连通至少有两种定义方式,这两种方式在定义局部连通空间时是一致的,但是在定义某一点的局部连通性时却是有差别的.本文将通过具体的例子指出其中的差异,并建立它们之间的联系.

There are at least two different ways of defining local connectedness.For locally connected space,these two types of definition are equivalent,while they are distinct for local connectedness at a certain point.We shall discriminate the difference through an example,and build their connections by proof.