Corollary 2

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a proper, separated and surjective mapping. Assume in addition one of the following hypotheses:

(i) The fibres of $f$, as well as the space $X\times_YX$, are locally connected;

(ii) The fibres of $f$ are finite, the diagonal $\Delta_X$ of $X\times_YX$ is open in $X\times_YX$ and the space $(X\times_YX)-\Delta_X$ is locally connected. 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 be a covering of X and let $\tau$ be a descent datum relative to $f$ on Z. By proposition 5 (IV, p. 389), the descent datum $\tau$ is effective, in other words the square

Z $Z/R_{\tau}$

X $^f$ Y

is a cartesian square. The hypotheses of proposition 6 (IV, p. 391) are then satisfied. Consequently, $Z/R_{\tau}$ is a covering of Y.