Proposition 19

If $H$ is a discrete normal subgroup of a topological group $G$, then $G/H$ is locally isomorphic to $G$.

Let $V$ be a neighbourhood of $e$ in $G$ which does not contain any point of $H$ other than $e$, and let $W$ be a symmetric open neighbourhood of $e$ in $G$ such that $W^2 \subset V$. Then the restriction to $W$ of the canonical mapping $\varphi$ of $G$ onto $G/H$ is injective; for if $x, y \in W$ and $\varphi(x) = \varphi(y)$, then $x^{-1}y \in W^2 \subset V$ and $x^{-1}y \in H$, so that $x = y$. By Proposition 17 it follows that the restriction of $\varphi$ to $W$ is a homeomorphism of $W$ onto $\varphi(W)$; since moreover $\varphi(xy) = \varphi(x)\varphi(y)$ for all $x, y \in W$, we conclude that $G$ and $G/H$ are locally isomorphic (§ 1, no. 3, Proposition 3).