Exercise 21, § 4

a) Let $X$ be a paracompact space and let $R$ be a closed equivalence relation on $X$ such that every equivalence class with respect to $R$ is compact. Show that $X/R$ is paracompact [use Exercise 15 a), Theorem 3 of no. 3, and Lemma 6 of no. 5; cf. Exercise 15 c)].

b) Let $X$ be a Hausdorff space and let $R$ be a closed equivalence relation on $X$ such that every equivalence class with respect to $R$ is compact. Show that if $X/R$ is paracompact, then so is $X$ (use the argument of Chapter I, $§ 9$, no. 10, Proposition 17).

c) Let $Y$ be a locally compact but not paracompact space (cf. Chapter I, $§ 9$, Exercise 12), and let $Y_0$ be the one-point compactification of $Y$. Let $X$ be the topological sum of $Y$ and $Y_0$, and let $R$ be the equivalence relation on $X$ which identifies each point of $Y$ with its canonical image in $Y_0$. The relation $R$ is open, every equivalence class mod $R$ contains at most two points, and $X/R$ is homeomorphic to $Y_0$, hence is compact.