Remark 1

Let $G$ be a non-Hausdorff topological group such that the Hausdorff group associated with $G$ is metrizable; equivalently, such that the identity element of $G$ has a countable fundamental system of neighbourhoods in G. The proof of Proposition 4 applies to this case without modification, H being an arbitrary normal subgroup of G, and shows that the Hausdorff group associated with G/H is metrizable, and that G/H is a complete group (in general not Hausdorff) whenever G is complete.