Exercise 31, § 4

a) Let $X$ be a paracompact space, $R$ a closed equivalence relation in $X$, such that every equivalence class under $R$ is compact. Prove that $X/R$ is paracompact. (It is known that $X/R$ is normal (IX, p. 105, exerc. 19). Use then IX, p. 107, exerc. 25 c)).¹ b) Let $X$ be a separated space, $R$ a closed equivalence relation in $X$, such that every equivalence class under $R$ is compact. Prove that if $X/R$ is paracompact, $X$ is paracompact (reason as in I, p. 70, prop. 17). c) Let $Y$ be a locally compact non-paracompact space (I, p. 106, exerc. 12), $Y_0$ the compact space obtained by adjoining to $Y$ a point at infinity. Let $X$ be the topological sum space of $Y$ and $Y_0$, $R$ the equivalence relation identifying every point of $Y$ with its canonical image in $Y_0$; the relation $R$ is open, every equivalence class under $R$ contains at most two points, and $X/R$ is homeomorphic to $Y_0$, hence compact.