$\pi$-metrizable spaces and strongly $\pi$-metrizable spaces

TL;DR

This paper investigates the properties and characterizations of $ ext{pi}$-metrizable and strongly $ ext{pi}$-metrizable spaces, providing new equivalences, preservation properties, and introducing related concepts.

Contribution

It offers new characterizations of $ ext{pi}$-metrizable spaces, proves preservation under certain maps, and introduces the concepts of second-countable and strongly $ ext{pi}$-metrizable spaces.

Findings

$ ext{pi}$-metrizability characterized by $ ext{sigma}$-hereditarily closure-preserving $ ext{pi}$-bases

Open and closed maps preserve $ ext{pi}$-metrizability

Introduces and studies second-countable and strongly $ ext{pi}$-metrizable spaces

Abstract

A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. In this paper, we mainly give affirmative answers for two questions about $\pi$-metrizable spaces. The main results are that: (1) A space $X$ is $\pi$-metrizable if and only if $X$ has a $\sigma$-hereditarily closure-preserving $\pi$-base; (2) $X$ is $\pi$-metrizable if and only if $X$ is almost $\sigma$-paracompact and locally $\pi$-metrizable; (3) Open and closed maps preserve $\pi$-metrizability; (4) $\pi$-metrizability satisfies hereditarily closure-preserving regular closed sum theorems. Moreover, we define the notions of second-countable $\pi$-metrizable and strongly $\pi$-metrizable spaces, and study some related questions. Some questions about strongly $\pi$-metrizability are posed.