Example 1

Let E be a topological space, G a discrete topological group operating continuously and freely in E on the right. In order that the canonical mapping $p: E\rightarrow E/G$ make E a principal covering of $E/G$, it is necessary and sufficient that it be étale (condition (iii) of proposition 4). For example, suppose there exists a topological space X and an étale mapping $q: E\rightarrow X$ compatible with the operation of G in E; then E is a principal covering of $E/G$. In fact, let us denote by $q': E/G\rightarrow X$ the mapping deduced from $q$; one has $q=q'\circ p$, therefore $p$ is étale by proposition 6, d) of I, p. 29.