Proposition 11

Let $G$ be a Hausdorff topological group operating properly on a non-empty space $X$. If $X$ is compact (resp. locally compact) then so are $G$ and $X/G$.

By hypothesis, the mapping $\theta : (s, x) \to (x, s.x)$ of $G \times X$ into $X \times X$ is proper; if $X \times X$ is compact (resp. locally compact) then the Corollary to Proposition 9 of Chapter I, $§ 10$, no. 4 shows that $G \times X$ is also compact (resp. locally compact), and therefore so is $G$ since $X \neq \emptyset$. Since $X/G$ is Hausdorff (no. 2, Proposition 3), the compactness (resp. local compactness) of $X$ implies the compactness (resp. local compactness) of $X/G$ [Chapter I, $§ 10$, no. 4, Proposition 8 (resp. Proposition 9)] (see $§ 2$, Exercise 29).

We shall now give criteria which allow us to assert that a Hausdorff topological group $G$ operates properly on a locally compact space $X$. For each pair of subsets $K, L$ of $X$ we denote by $P(K, L)$ the set of all $s \in G$ such that $s.K \cap L \neq \emptyset$.