Regular Bases At Non-isolated Points And Metrization Theorems

Details Regular Bases At Non-isolated Points And Metrization Theorems Regular Bases At Non-isolated Points And Metrization Theorems

2011-06-20

arxiv.org/abs/1106.3808v1

Mathematics

General Topology

12 pages

Scientific paper

In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space $X$ is a metrizable space, if and only if $X$ is a regular space with a $\sigma$-locally finite base at non-isolated points, if and only if $X$ is a perfect space with a regular base at non-isolated points, if and only if $X$ is a $\beta$-space with a regular base at non-isolated points. In addition, we also discuss the relations between the spaces with a regular base at non-isolated points and some generalized metrizable spaces. Finally, we give an affirmative answer for a question posed by F. C. Lin and S. Lin in \cite{LL}, which also shows that a space with a regular base at non-isolated points has a point-countable base.

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