Corollary 3

Let B be a topological space, let $(E, p)$ and $(E', p')$ be B-spaces and let $f: E'\rightarrow E$ be a B-morphism. Suppose that $E'$ is a locally finite covering of B.

a) If the mapping $p$ is étale and separated, the E-space $(E', f)$ is a covering.

b) If the mapping $f$ is surjective and makes the space $E'$ a covering of E, then E is a covering of B.

Under assumption a), the mappings $p$ and $p'$ are etale and separated, and the mapping $p'$ is proper (theorem 1). The mapping $f$ is therefore etale (I, p. 30, cor. 1 of prop. 6), separated (I, p. 25, prop. 2 b)) and proper (I, p. 28, prop. 5). By theorem 1, $(E', f)$ is a covering of E.

Now suppose that $f$ is surjective and that $(E', f)$ is a covering of E. Then it is locally finite, hence the mapping $f$ is proper and surjective (th. 1). By I, p. 25, prop. 2, c), the mapping $p$ is separated, since $p'=p\circ f$ and $p'$ is separated. The mapping $p$ is etale (I, p. 29, prop. 6, d)), proper (TG, I, p. 73, prop. 5, b)) and its fibres are finite. By th. 1, the B-space $(E, p)$ is then a covering of B.