Remark

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.

a) If E and $E'$ are coverings of B, the mapping $f$ is etale (I, p. 29, prop. 6) and separated (I, p. 25, prop. 2). But it is not true, in general, that $(E', f)$ is a covering of E if the space B is not locally connected (I, p. 145, exerc. 4).

b) If $(E', p')$ is a covering of B, $f$ is surjective and makes $E'$ a covering of E, then the mapping $p$ is etale (I, p. 29, prop. 6). But it is not true, in general, that it is separated if the space B is not locally connected (I, p. 145, exerc. 4). In particular, E is not necessarily a covering of B.