Exercise 7, § 10

Let $f : X \to X'$ be a proper mapping and let $R$ (resp. $R'$) be an equivalence relation on $X$ (resp. $X'$) with which $f$ is compatible. Let $\overline{f} : X/R \to X'/R'$ be the mapping of the quotient spaces induced by $f$. Suppose that (i) the image under $f$ of every saturated set with respect to $R$ is saturated with respect to $R'$, and (ii) $R'$ is open. Show that in these conditions $\overline{f}$ is proper [use Theorem 1 c)].