Theorem 20.2:

Let $Y$ be a Hausdorff space and $G$ be a group of acting properly discontinuously on $Y$ as a group of homeomorphisms. Then,

(i) The orbit space $Y/G$ is Hausdorff.

(ii) The quotient map $${ \eta : Y \longrightarrow Y/G }$$ is a covering projection.

(iii) $G$ is the group of deck-transformations for the covering projection $\eta : Y \longrightarrow Y/G$.

(iv) In case $Y$ is simply connected, $\pi_1(Y/G)$ is isomorphic to $G$.