Proposition 7

Let $G$ be a locally compact group operating continuously on a Hausdorff space $X$. Then $G$ operates properly on $X$ if and only if, for each pair of points $x, y$ of $X$, there is a neighbourhood $V_x$ of $x$ and a neighbourhood $V_y$ of $y$ such that the set $K$ of all $s \in G$ for which $s.V_x \cap V_y \neq \emptyset$ is relatively compact in $G$.

Let $F$ be the compact space obtained by adjoining a point at infinity $\omega$ to $G$, and let $\Gamma$ be the graph of $\rho : (s, x) \to s.x$ considered as a subset of $F \times X \times X$. Let us show that if the restriction of $\mathrm{pr}_{23}$ to $\Gamma$ is proper, then $\Gamma$ is closed in $F \times X \times X$. Indeed, this hypothesis implies that the map $u : (t, s, x, y) \to (t, x, y)$ of $F \times \Gamma$ into $F \times X \times X$ is closed. If $\Gamma'$ is the set of points $(s, s)$ in $F \times G$, where $s \in G$, then $\Gamma'$ is closed in $F \times G$, since it is the graph of the canonical injection $G \to F$ (Chapter I, § 8, no. 1, Corollary 2 to Proposition 2); hence the intersection $(\Gamma' \times X \times X) \cap (F \times \Gamma)$ is closed in $F \times \Gamma$, and it is immediately seen that its image under $u$ is the set $\Gamma$ considered as a subset of $F \times X \times X$; hence $\Gamma$ is closed in $F \times X \times X$. Now, we have $(\{\omega\} \times X \times X) \cap \Gamma = \emptyset$. From the definition of $F$, therefore, for every point $(x, y) \in X \times X$ there is a neighbourhood $W$ of $(x, y)$ in $X \times X$ and a compact subset $K$ of $G$ such that

is empty; and since we may take $W$ to be a neighbourhood $V_x \times V_y$, where $V_x$ and $V_y$ are neighbourhoods of $x$ and $y$ respectively in $X$, the statement “((G — K) × W) ∩ Γ = ∅” becomes “if s ∉ K, then s · V_x ∩ V_y = ∅”. We have thus proved the necessity of the condition stated in the proposition. Conversely, suppose this condition is satisfied; let A be a set filtered by an ultrafilter $\mathfrak{F}$, and let α → (s_α, x_α) be a mapping of A into G × X such that lim_$\mathfrak{F}$ x_α = x and lim_$\mathfrak{F}$ s_α · x_α = y. Suppose that K, V_x and V_y satisfy the condition of the proposition. By hypothesis there is a set M ∈ $\mathfrak{F}$ such that if α ∈ M then x_α ∈ V_x and s_α · x_α ∈ V_y, hence s_α ∈ K. This shows that α → s_α converges with respect to $\mathfrak{F}$, and the proof is complete.

If G is compact, the condition of Proposition 7 is trivially satisfied; we thus retrieve Proposition 2 a).

Proposition 7 shows in particular that a discrete group G, operating continuously on a Hausdorff space X, operates properly on X if and only if, for each pair (x, y) of points of X, there is a neighbourhood V_x of x and a neighbourhood V_y of y such that the set of points s ∈ G for which s · V_x ∩ V_y ≠ ∅ is finite.