Regular Bases At Non-isolated Points And Metrization Theorems

TL;DR

This paper characterizes metrizable spaces using regular bases at non-isolated points and establishes their relation to generalized metrizable spaces, providing new metrization criteria and answering an open question.

Contribution

It introduces the concept of regular bases at non-isolated points and proves new metrization theorems involving these bases, expanding understanding of generalized metrizable spaces.

Findings

Characterization of metrizable spaces via regular bases at non-isolated points

Equivalence of various conditions involving regular bases and perfect spaces

Spaces with a regular base at non-isolated points have a point-countable base

Abstract

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.