Proposition 7

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a continuous and surjective mapping. Suppose that the space Y is unloopable and that the mapping $f$ is open and has the path-lifting property. Then every descent datum relative to $f$ on a covering of X is effective, and the quotient space is a covering of Y.

Let $(Z, p)$ be a covering of X and let $\tau$ be a descent datum relative to $f$ on Z. Since the mapping $f$ is surjective and open, it follows from prop. 5 of IV, p. 389, that the descent datum $\tau$ is effective. Let $T = Y/R_{\tau}$; it is the quotient Y-space; its projection $q$ is étale (loc. cit.); it is also separated (I, p. 27, prop. 4).

By hypothesis, the mapping $f$ has the path-lifting property, and so does the mapping $p$, since Z is a covering of X (III, p. 302, Corollary 2 of Prop. 3). Hence the same is true of the mapping $p\circ f$, and therefore of the mapping $q$. Consequently (cf. IV, p. 341, Remark 2), T is a covering of Y. The proposition is thus proved.