# Free cocompact action of discrete group gives a covering map

I’d like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah’s paper “Elliptic operators, discrete groups, and von Neumann algebras.” http://www.maths.ed.ac.uk/~aar/papers/atiyah_elld.pdf

Suppose a discrete group $G$ acts freely on a manifold $X$ with the quotient $X/G$ being compact. Then $X$ is a covering space of $X/G$ with covering map given by the quotient map $p: X\rightarrow X/G$.

Thanks.

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The discreteness assumption is meaningless since given a continuous group action $G\times X\to X$, it remains continuous if we equip $G$ with discrete topology. Now, a counter-example to the claim is the action of the group of the additive group of real numbers on itself
$${\mathbb R}_{d} \times {\mathbb R} \to {\mathbb R}$$
via addition. Here the subscript $d$ means the discrete topology, otherwise ${\mathbb R}$ is equipped with the standard topology. The quotient space is a single point, hence, compact. The action itself is clearly not a covering action, but is free. I think, Atiyah forgot to add proper, in the sense of Palais:

A continuous action $G\times X\to X$ is proper, in the sense of Palais, if for any two points $x, y\in X$ there exists a pair of neighborhoods $U_x, U_y$ of these points such that
$$\{g\in G: g U_x\cap U_y\ne \emptyset\}$$
is a relatively compact subset of $G$. When $G$ is discrete, this just means that this subset is finite.

The correct statement is then:

If $G\times X\to X$ is a Palais-proper, free, action of a discrete group on a Hausdorff space, then the projection X\to X/G\$ is a covering map.

Related: Proper and free action of a discrete group.

Dave Glickenstein’s notes give a short proof, see page 9.