Corollary 1

A Hausdorff uniformity defined by a countable family of pseudometrics is metrizable.

For if $(f_n)$ is a sequence of pseudometrics defining such a structure, the filter of entourages is generated by the countable family of sets $\overline{f}_n^1([0, 1/m])$, where $m$ and $n$ each run through the set of integers $> 0$.