Proposition 2

Let $f: X\rightarrow$ Y be a continuous mapping, let $(Z, p)$ be an X-space and let $\tau$ be a descent datum relative to $f$ on Z. The equivalence relation $R_{\tau}$ is closed if $f$ is proper; it is open if $f$ is open.

The mapping $\widetilde{\tau}: Z\times_YX\rightarrow$ X $\times_Y$ Z defined by $(z, x)\mapsto$ $(p(z), \tau (z, x))$ is a homeomorphism, with inverse mapping $(x, z)\mapsto (\tau (z, x), p(z))$. We have $\tau$ = pr$_2\circ \widetilde{\tau}$, where pr$_2: X\times_YZ\rightarrow$ Z is the second projection. If $f$ is proper, pr$_2$ is proper; if $f$ is open, pr$_2$ is open (I, p. 17, prop. 8). It follows that $\tau$ is proper (resp. open) if $f$ is. The saturation of a subset A of Z with respect to the relation $R_{\tau}$ is the image of $A\times_YX$ under $\tau$. Consequently, if $f$ is proper, the saturation of a closed subset is closed; if $f$ is open, the saturation of an open subset is open.