Properly discontinuous action

Remarks

When $X$ is Hausdorff and locally compact, the action is properly discontinuous if and only if the action of $\Gamma$ with the discrete topology is proper. In covering space theory, some authors use the term for the weaker condition that each point has a neighbourhood $U$ with $\gamma U \cap U = \emptyset$ for all $\gamma \ne 1$.

Example: let $\Z^n$ act on $\R^n$ by translations. If $K$ and $L$ lie in the ball of radius $R$ about $0$, then $(v + K) \cap L \ne \emptyset$ forces $\norm{v} \le 2R$. Only finitely many $v \in \Z^n$ satisfy this, so the action is properly discontinuous.

Non-example: let $\Z$ act on $\R$ by $n \cdot x = 2^n x$. For $K = L = \set{0}$, every $n \in \Z$ satisfies $(n \cdot K) \cap L \ne \emptyset$, so the set is infinite. So the action is not properly discontinuous.