Proposition 13

Let G be a locally compact group and let H be a closed subgroup of G. Then the homogeneous space $G/H$ is locally compact and paracompact.

Since $G/H$ is Hausdorff (§ 2, no. 5, Proposition 13) it is locally compact, by Proposition 9 of no. 5 applied to H operating on G on the right. Thus it remains to show that $G/H$ is paracompact. Let V be a symmetric compact neighbourhood of e in G, and let $G_0 = V^\infty$ be the subgroup of G generated by V. $G_0$ is open (§ 2, no. 1, Corollary to Proposition 4) and operates continuously on $G/H$ (§ 2, no. 5, Proposition 12). If we can show that each of the orbits $G_0.z \ (z \in G/H)$ is an open subset of $G/H$ and a countable union of compact sets, then it will follow that $G/H$ is the topological sum of the distinct orbits $G_0.z$ and is therefore paracompact (Chapter I, , no. 10, Theorem 5). That $G_0 . z$ is open in $G/H$ follows from the facts that $G_0$ is open in $G$ and the equivalence relation $x^{-1} y \in H$ is open ($\S 2$, no. 4, Lemma 2). On the other hand, $G_0 z$ is the union of the sets $V^n . z$ ($n \geq 1$), and since $V^n$ is compact in $G$ and $G/H$ is Hausdorff, $V^n . z$ is compact. This completes the proof