Proposition 3

Let B be an unloopable topological space. Let $(Y, q)$ be a covering of B and let $(Z, p)$ be a covering of Y. The topological space Z, endowed with the mapping $q\circ p$, is then a covering of B.

In fact, the mapping $q\circ p$ is étale (I, p. 29, prop. 6) and separated (I, p. 25, prop. 2). By Remark 2 above, it is therefore enough to prove that it satisfies the path-lifting property. Let $z$ be a point of Z and let $c:\mathbf{I}\rightarrow B$ be a path in B with origin $q(p(z))$. There exists a path $c'$ with origin $p(z)$ in Y which lifts $c$, because Y is a covering of B (III, p. 302, Cor. 2). Since Z is a covering of Y, there exists a path $c''$ with origin $z$ in Z which lifts $c'$; the path $c''$ lifts $c$. This proves the proposition.