Proposition 12

Let G be a Hausdorff topological group which operates continuously on a topological space X. Let K be a compact subset of X, and let $\rho_K$ be the mapping $(s, x) \to s.x$ of $G \times K$ into X. Then:

a) If G operates properly on X, $\rho_K$ is proper.

b) If X is locally compact and $\rho_K$ is proper for each compact subset K of X, G operates properly on X.

The mapping $\rho_K$ factorizes as $G \times K \xrightarrow{\theta_K} K \times X \xrightarrow{\mathrm{pr}_2} X$, where $\theta_K$ is the restriction to $G \times K$ of the mapping $\theta : (s, x) \to (x, s.x)$ of $G \times X$ into $X \times X$. Since $\theta^{-1}(K \times X) = G \times K$, $\theta_K$ is proper if $\theta$ is proper (Chapter I, § 10, no. 1, Proposition 3). On the other hand, since K is compact, the projection $\mathrm{pr}_2 : K \times X \to X$ is proper (Chapter I, § 10, no. 2, Theorem 1, Corollary 5), hence $\rho_K$ is proper (ibid., no. 1, Proposition 5).

Suppose conversely that $\rho_K$ is proper for each compact $K \subset X$. If L is a compact subset of X, then $\rho_K^{-1}(L)$ is a compact subset of $G \times K$, whose projection into G is $P(K, L)$; therefore $P(K, L)$ is compact. Hence if X is locally compact it follows from Theorem 1 that G operates properly on X.