On symmetrizability and perfectness of second-countable spaces

TL;DR

This paper explores conditions under which second-countable Hausdorff spaces are symmetrizable, establishing a link with perfectness, and provides examples and cardinality bounds for non-symmetrizable spaces.

Contribution

It proves a symmetrizability criterion for second-countable Hausdorff spaces and constructs examples of non-symmetrizable spaces with minimal cardinalities.

Findings

A second-countable Hausdorff space is symmetrizable iff it is perfect.

An example of a non-symmetrizable second-countable submetrizable space of cardinality .

Determination of the minimal cardinality  of non-symmetrizable second-countable T_i-spaces.

Abstract

A symmetrizability criterion of Arhangelskii implies that a second-countable Hausdorff space is symmetrizable if and only if it is perfect. We present an example of a non-symmetrizable second-countable submetrizable space of cardinality $\mathfrak q_0$ and study the smallest possible cardinality $\mathfrak q_i$ of a non-symmetrizable second-countable $T_i$-space for $i\in\{1,2\}$.