Example 2

Let us take up again the notation of Example 2 (IV, p. 383). The topological space $Z/R_{\tau}$ is then the topological space obtained by gluing the spaces $Z_i$ along the $\overset{-1}{p_{i}}(V_i\cap V_j)$ by means of the bijections $\tau_{i,j}$ (TG, I, p. 16). Consequently, if for every $i\in I$, the set $V_i$ is open (resp. closed) in Y, the set $g(Z_i)$ is open (resp. closed) in $Z/R_{\tau}$ and the restriction of $g$ to $Z_i$ induces a homeomorphism of $Z_i$ onto $g(Z_i)$ (TG, I, p. 17, Prop. 9). The space Z is the sum space of the spaces $Z_i$; the space $X\times_Y(Z/R_{\tau})$ is the sum space of the spaces $V_i\times_Y$ $(Z/R_{\tau}) =g(Z_i)$. The mapping $h$ is identified with the sum mapping of the mappings $g|Z_i: Z_i\rightarrow g(Z_i)$. It is therefore a homeomorphism, which shows that the descent datum $\tau$ is effective.

Without a particular assumption on the subsets $V_i$, it is not always true that the restriction of $g$ to $Z_i$ induces a homeomorphism of $Z_i$ onto its image; in this case, the descent datum $\tau$ is not effective (IV, p. 462, exerc. 2).