Remark

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a mapping. In order that every descent datum relative to $f$ on a covering of X be effective, and that the quotient space be a covering of Y, it is necessary that $f$ be strict and that $f(X)$ be an open and closed subset of X.

In fact, let us identify the X-space X with $X\times_YY$ and endow it with its canonical descent datum relative to $f$. The quotient space is identified with $f(X)$, endowed with the quotient topology of the topology of X for the equivalence relation defined by $f$. If this is a covering of Y, the space $f(X)$ is then identified with an open and closed subset of Y and the mapping $f$ is strict.