Theorem 1

Let G be a discrete group operating continuously on the right in a topological space E. Suppose that every point of E possesses a neighbourhood U such that $U\cap (U\cdot g) =\emptyset$ for every element $g$ of G other than the identity element. Then the canonical mapping $p: E\rightarrow E/G$ makes E a principal covering of $E/G$ with group G.

The operation of G in E is free by assumption, so it is enough to prove that the mapping $p$ is étale (I, p. 98, example 1). Let $x$ be a point of E and let U be an open neighbourhood of $x$ such that $U\cap U\cdot g=\emptyset$ for every element $g$ of G distinct from the identity element. The mapping $p$ is open (TG, III, p. 10, lemma 2) and induces an injective, continuous and open mapping of U onto $p(U)$, hence a homeomorphism, which proves that the mapping $p$ is étale.