Theorem 1

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 $L$ be a closed subset of $X$. Then: a) The set $P(K, L)$ is closed in $G$.

b) If G operates properly on X and if L is compact, then P(K, L) is compact.

c) Conversely, if X is locally compact and if, for each pair K, L of compact subsets of X, P(K, L) is relatively compact in G [and therefore compact by a)]; then G operates properly on X (and if X is not empty, G is locally compact by Proposition 11).

The mapping $(s, x) \to s.x$ of $G \times K$ into $X$ is continuous, and the inverse image $L'$ of $L$ under this mapping is therefore closed. Since $K$ is compact, the projection $\operatorname{pr}_1 : G \times K \to G$ is proper (Chapter I, § 10, no. 2, Corollary 5 of Theorem 1) and the image of $L'$ under $\operatorname{pr}_1$ is therefore closed. This image is $P(K, L)$, hence a) is proved.

To prove b): $X$ is Hausdorff (no. 2, Proposition 3). By hypothesis the mapping $\theta : (s, x) \to (x, s.x)$ of $G \times X$ into $X \times X$ is proper; since $K \times L$ is compact, so is $\overline{\theta}(K \times L)$ since $X$ is Hausdorff (Chapter I, § 10, no. 2, Proposition 6). Hence the projection $P(K, L)$ of $\overline{\theta}(K \times L)$ into $G$ is a compact set.

To prove c): since $K \times L$ is closed in $X \times X$, it follows that $\overline{\theta}(K \times L)$ is closed in $P(K, L) \times K$ and is therefore compact under the hypotheses of c). Since every compact subset of $X \times X$ is contained in a compact subset of the form $K \times L$, it follows that the inverse image of any compact subset of $X \times X$ under $\theta$ is compact, and since $X \times X$ is locally compact, this shows that $\theta$ is proper (Chapter I, § 10, no. 3, Proposition 7) [see Chapter IV, § 1, Exercise 4c)].