Proposition 1

The left and right uniformities of a topological group $G$ are metrizable if and only if $G$ is Hausdorff and the identity element $e$ of $G$ has a countable fundamental system of neighbourhoods.

The condition is clearly necessary. Conversely, if it is satisfied, let $(V_n)$ be a fundamental system of neighbourhoods of $e$; if $U_n$ denotes the set of pairs $(x, y) \in G \times G$ such that $x^{-1}y \in V_n$, then the $U_n$ form a countable fundamental system of entourages of the left uniformity of $G$; since this uniformity is Hausdorff, it follows from § 2, no. 4, Theorem 1 that it is metrizable. Similarly for the right uniformity of $G$.

A topological group $G$ is said to be metrizable if its topology is metrizable. Proposition 1 then shows that its two uniformities are metrizable.

This result can be sharpened with the help of the following notion: