Nagata–Smirnov metrization theorem

Remarks

For a metric space, Stone's theorem gives, for each $n$, a locally finite open refinement $\mathcal{B}_n$ of the cover by balls of radius $1/n$. The union of the $\mathcal{B}_n$ is a $\sigma$-locally finite base. For the converse one shows that $X$ is normal and builds countably many locally finite families of continuous functions to $[0,1]$. These functions embed $X$ into a metrizable product space. A countable base is $\sigma$-locally finite, since each one-element family is locally finite. So the Urysohn metrization theorem is a special case.

Going deeper

For a student

Together with the Bing metrization theorem, this is the general solution of the metrization problem. Its proof uses Urysohn's lemma to build the continuous functions of the embedding. A useful consequence is that a paracompact Hausdorff space that is locally metrizable is metrizable.