Proposition 6

Let $G$ be a topological group operating continuously on a topological space $X$, and suppose that $G$ operates freely on $X$. Then $G$ operates properly on $X$ if and only if the following condition is satisfied:

(FP) The graph $C$ of the equivalence relation defined by $G$ is closed in $X \times X$, and the canonical mapping $\varphi : C \to G$ is continuous.

The set $C$ is the image of the mapping $\theta : (s, x) \to (x, s.x)$ of $G \times X$ into $X \times X$. We know (Chapter I, § 10, no. 1, Proposition 2) that $\theta$ is proper if and only if $C$ is closed in $X \times X$ and (if $\theta'$ denotes the mapping $\theta$ considered as a mapping of $G \times X$ into $C$) $\theta'$ is a homeomorphism. Now the hypothesis implies that $\theta'$ is bijective and that its inverse is the mapping $(x, y) \to (\varphi(x, y), x)$. Hence $\theta'$ is a homeomorphism if and only if $\varphi$ is continuous.