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 σ-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 β-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 [7], which also shows that a space with a regular base at non-isolated points has a point-countable base.
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