Exercise 2, § 4

Soit Y a topological space obtained by pasting together spaces $X_i$; let X be the topological sum of the family $(X_i)$ and let $f: X\rightarrow Y$ be the canonical mapping (TG, I, p. 16). For every $i$, put $Y_i=f(X_i)$ and let $f_i: X_i\rightarrow Y_i$ be the mapping deduced from $f$ by passing to the subspaces. The mapping $f_i$ is bijective and continuous. If the mapping $f_i$ is not a homeomorphism, the canonical descent datum on X, relative to $f$, is not effective. For an example, cf. TG, I, p. 94, exer. 15.