Corollary 1

Let G be a discrete topological group operating continuously on the right in a topological space E. The following conditions are equivalent:

(i) The group G operates properly and freely in E;

(ii) The space $E/G$ is separated and the space E is a principal covering with base $E/G$ and group G.

Moreover, under these conditions, the space E is separated.

Suppose condition (i) satisfied. Then the spaces E and $E/G$ are separated (TG, III, p. 29, prop. 3) and for every $x\in E$ there exists an open neighbourhood U of $x$ in E such that $U\cap (U\cdot g) =\emptyset$ for every element $g$ of G other than the identity element (TG, III, p. 32, prop. 8). By Theorem 1, condition (ii) is then satisfied.

The implication (ii)$\Rightarrow$(i) follows from corollary 1 of I, p. 96.