Proposition 13

Let G be a topological group and let H be a subgroup of G. Then the homogeneous space $G/H$ is Hausdorff if and only if H is closed in G.

H is an equivalence class for the relation $x^{-1}y \in H$ and therefore, if $G/H$ is Hausdorff, H is closed in G. Conversely, if H is closed, then the graph of this relation is closed in $G \times G$, since it is the inverse image of H under the continuous mapping $(x, y) \to x^{-1}y$. Since the relation $x^{-1}y \in H$ is open, it follows from Chapter I, § 8, no. 3, Proposition 8 that $G/H$ is Hausdorff.