Weak extent, submetrizability and diagonal degrees

TL;DR

This paper explores how certain diagonal properties and the countable weak extent of a space's square influence its submetrizability and condensation onto second countable spaces, extending previous results.

Contribution

It generalizes earlier theorems by establishing new conditions under which spaces with specific diagonal properties are submetrizable or condense onto second countable spaces.

Findings

Spaces with zero-set diagonals and countable weak extent are submetrizable.

Spaces with regular G_delta-diagonals and countable weak extent condense onto second countable spaces.

Cardinality bounds are established based on diagonal degrees.

Abstract

We show that if $X$ has a zero-set diagonal and $X^2$ has countable weak extent, then $X$ is submetrizable. This generalizes earlier results from Martin and Buzyakova. Furthermore we show that if $X$ has a regular $G_\delta$-diagonal and $X^2$ has countable weak extent, then $X$ condenses onto a second countable Hausdorff space. We also prove several cardinality bounds involving various types of diagonal degree.