Definition 1

Let $G$ be a topological group operating continuously on a topological space $X$. $G$ is said to operate properly on $X$ if the mapping

is proper (Chapter I, § 10, no. 1, Definition 1).

Let $\Gamma \subset G \times X \times X$ be the graph of the mapping $\rho : (s, x) \to s.x$. Since $\rho$ is continuous, the mapping $\sigma : (s, x) \to (s, x, s.x)$ is a homeomorphism of $G \times X$ onto $\Gamma$, and the composite mapping

is just $\theta$. Definition 1 is therefore equivalent to saying that the restriction of $\mathrm{pr}_{23}$ to $\Gamma$ is a proper mapping of $\Gamma$ into $X \times X$.

Theorem 1 of Chapter I, § 10, no. 2 shows that G operates properly on X if and only if the following condition is satisfied: For each set A filtered by an ultrafilter $\mathfrak{F}$, and each mapping

of A into $G \times X$, if the mapping $\alpha \to (s_\alpha \cdot x_\alpha, x_\alpha)$ has a limit $(b, a)$ with respect to $\mathfrak{F}$, then $\alpha \to s_\alpha$ has a limit $t \in G$ with respect to $\mathfrak{F}$, such that $t.a = b$.