Corollary

Let B be a connected topological space, locally connected by arcs, let $(E, p)$ be a B-space whose projection $p$ is etale, separated, and has the path-lifting property. If the canonical operation of $\varpi (B)$ on the set E, relative to $p$, is without local monodromy, then E is a covering of B.

Let us endow E with the canonical operation of $\varpi (B)$ defined by path lifting (III, p. 303, n$^o3$). There exists a topology on E for which conditions (i) and (ii) of prop. 3 are satisfied. This topology coincides with the given topology on E by lemma 1 of III, p. 312.