Proposition 18

*Let $G$ be a topological group and let $H$ be a normal subgroup of $G$. a) *The quotient group $G/H$ is Hausdorff if and only if $H$ is closed in $G$. b) The quotient group $G/H$ is discrete if and only if $H$ is open in $G$.

If $G$ is a topological group and $N$ is the closure of $\{e\}$ in $G$, then $N$ is a closed normal subgroup of $G$ (no. 1, Proposition 1), hence $G/N$ is Hausdorff; $G/N$ is called the Hausdorff group associated with $G$.