For questions pertaining to the metrizability of topological spaces and / or metrization theorems.
We call a topology on a set $X$ metrizable when we can endow $X$ with a metric that induces this topology. Determining whether a space is metrizable is, historically, a major problem in point-set topology, and much work has been dedicated to finding relevant necessary and/or sufficient conditions with results such as Urysohn's metrization theorem, the Nagata–Smirnov metrization theorem, et al.
