Proposition 7

Let B be a topological space, let E and $E'$ be B-spaces, and let $f: E'\rightarrow E$ be a B-morphism. It is assumed that $E'$ is a covering of B and that the space B is locally connected.

a) If the projection of the B-space E is an étale and separated mapping, $(E', f)$ is a covering of E.

b) If $f$ is surjective, then E is a covering of B.

Let $p$ and $p'$ denote the projections of the B-spaces E and $E'$ respectively. Under assumption a$),f$ is étale (I, p. 29, Prop. 6, c)). Under assumption b$),p$ is étale (loc. cit., d)). Hence, under either one of these two assumptions, E is locally connected; in particular, its connected components are open and closed in E (TG, I, p. 85, Prop. 11).

We shall first prove Proposition 7 under the additional assumption that B is connected, locally connected, and that $E'$ is the trivial covering $(B\times F'$, pr$_1)$.