Let $G$ be a group and $E$ a topological space on which $G$ acts by homeomorphisms, that is there is a homomorphism $\Gr\co G\to \Au(E)$ , where the latter denotes the group of self-homeomorphisms of $E$ . The action is said to be properly discontinuous if each point $e\in E$ has a neighborhood $U$ with the property that all non trivial elements of $G$ move $U$ outside itself: $$\forall g\in G\quad g \neq \id\Rightarrow gU\cap U=\emptyset\,.$$

For example, let $p\co E\to X$ be a covering map, then the group of deck transformations of $p$ acts properly discontinuously on $E$ . Indeed if $e\in E$ and $D\in \Au(p)$ then one can take as $U$ to be any neighborhood with the property that $p(U)$ is evenly covered. The following shows that this is the only example: