properly discontinuous action

Let $G$ be a group and $E$ a topological space on which $G$ acts byhomeomorphisms, that is there is a homomorphism $\rho :G\to \mathrm{Aut}(E)$, wherethe latter denotes the group of self-homeomorphisms of $E$. The action issaid to be properly discontinuous if each point $e\in E$ has aneighborhood $U$ with the property that all non trivial elements of $G$ move$U$ outside itself:

$$\forall g\in G\mathit{\hspace{1em}}g\ne \mathrm{id}\Rightarrow gU\cap U=\mathrm{\varnothing }.$$

For example, let $p:E\to X$ be a covering map, then the group of decktransformations of $p$ acts properly discontinuously on $E$. Indeed if $e\in E$ and $D\in \mathrm{Aut}(p)$ then one can take as $U$ to be any neighborhood withthe property that $p(U)$ is evenly covered. The following shows that thisis the only example:

Theorem.

Assume that $E$ is a connected and locally path connected Hausdorffspace. If the group $G$ acts properly discontinuously on $E$ then thequotient map $p\mathrm{:}E\mathrm{\to }E\mathrm{/}G$ is a covering map and $\mathrm{Aut}\mathit{}\mathrm{(}p\mathrm{)}\mathrm{=}G$.