Proposition 6

Let B be a topological space and let E be a B-space; let us denote by $p$ its projection. Suppose that the space E is locally connected. In order that E be a covering of B, it is necessary and sufficient that every point of B possess an open neighbourhood U such that the mapping $p$ induce a homeomorphism of every connected component of $\overset{-1}{p}(U)$ onto U.

Let $b$ be a point of B. If E is a covering of B, every open neighbourhood U of $b$ which is connected and over which the covering E is trivializable satisfies the conditions stated in the proposition. Conversely, let U be an open neighbourhood of $b$ satisfying these conditions. The set $\overset{-1}{p}(U)$ is open; its connected components are open subsets of E (TG, I, p. 85, prop. 11) and constitute a partition of $\overset{-1}{p}(U)$. The proposition therefore follows from corollary 1 (I, p. 70) to proposition 1.