Proposition 1

Let X and Y be topological spaces, let $f: X\rightarrow Y$ be a continuous mapping. Suppose that every point of Y possesses a neighbourhood above which there exists a continuous section of the mapping $f$. Then every descent datum relative to $f$ on an X-space is effective.

Let $(Z, p)$ be an X-space and let $\tau$ be a descent datum relative to $f$ on $(Z, p)$. The assertion that $\tau$ is an effective descent datum is local in Y, which allows us to suppose that the mapping $f$ possesses a continuous section $s$. Let $g: Z\rightarrow Z/R_{\tau}$ and $q: Z/R_{\tau}\rightarrow Y$ denote the canonical mappings, and let $h: Z\rightarrow X\times_Y(Z/R_{\tau})$ be the mapping given by $z\mapsto (p(z), g(z))$. The mapping $h$ is bijective and continuous; it is enough to prove that it is a homeomorphism.

The mapping from Z into Z which, to $z$, associates $\tau (z, s(f(p(z))))$ is continuous and maps every element of Z to the unique element $z'$ of Z which is equivalent to it for the relation $R_{\tau}$ and such that $p(z')$ belongs to the image of $s$. It therefore defines, by passing to the quotient, a continuous mapping $t: Z/R_{\tau}\rightarrow Z$ which is a section of the mapping $g$. In particular, one has $f\circ p\circ t=q\circ g\circ t=q$.

For every $(x, u)\in X\times_Y(Z/R_{\tau})$, one has $f(p(t(u))) =f(x)$; let us then put $h'(x, u) =\tau (t(u), x)$. The resulting mapping $h'$ from $X\times_Y(Z/R_{\tau})$ into Z is continuous. For every $(x, u)\in X\times_Y(Z/R_{\tau})$, one has

because $\tau (t(u), x)$ is equivalent to $t(u)$ for the relation $R_{\tau}$. This proves that the mapping $h\circ h'$ is the identity mapping of $X\times_Y(Z/R_{\tau})$. For $z\in Z$, one then has $t(g(z)) =\tau (z, s(f(p(z)))$ and $f(p(t(g(z)))) =f(p(z))$, whence $z=\tau (t(g(z)), p(z))$, by definition of the equivalence relation $R_{\tau}$. Thus one has $h'(h(z)) =z$ and $h'\circ h=$ Id$_Z$. The mapping $h$ is therefore a homeomorphism, which was to be proved.