Weak extent, submetrizability and 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.