I'd like to find a short proof of the following seemingly basic fact. 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 $p: X\rightarrow X/G$.
This appears 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
Thanks.