Proper group action

Remarks

When $G$ and $X$ are Hausdorff and locally compact, $\theta$ is proper if and only if the preimage of every compact set is compact. Then the action is proper if and only if, for every compact $K \subseteq X$, the set $\set{g \in G : (g \cdot K) \cap K \ne \emptyset}$ is compact.

Example: let $G = \Z$ act on $X = \R$ by $n \cdot x = x + n$. For compact $K \subseteq [-R, R]$, the condition $(n + K) \cap K \ne \emptyset$ forces $\abs{n} \le 2R$, so the set of such $n$ is finite, hence compact in $\Z$. So the action is proper.

Non-example: let $G = (0, \infty)$ under multiplication act on $X = \R$ by $t \cdot x = tx$. For $K = [-1, 1]$, the point $0$ lies in $t \cdot K \cap K$ for every $t$, so the set of such $t$ is all of $(0, \infty)$, which is not compact. So the action is not proper.

Every continuous action of a compact Hausdorff group on a Hausdorff space is proper.