Example 2

Let $q: E\rightarrow$ X be an étale and separated mapping. The group Aut$_X(E)$, endowed with the discrete topology, operates continuously on the left in E. If the space E is connected, this operation is free (I, p. 34, corollary 2). Let us denote by G the group Aut$_X(E)^{\circ}$ and endow E with the right operation opposite to that of Aut$_X(E)$. By the preceding example, the space E, endowed with the canonical mapping $p: E\rightarrow E/G$, is a principal covering with group G.