Exercise 15, § 4

a) Let $X$ be a normal space and let $R$ be a closed equivalence relation on $X$. Show that the quotient space $X/R$ is normal (use Proposition 10 of Chapter I, § 5, no. 4).

b) Let $X$ be a locally compact $\sigma$-compact space, and let $R$ be an equivalence relation on $X$ whose graph is closed in $X \times X$. Show that $X/R$ is normal (cf. Chapter I, § 10, Exercise 19).

c) Let $X$ be the complement in $\mathbf{R}$ of the set of points of the form $1/n$, where $n$ is any integer except 0 or $\pm 1$. Consider the equivalence relation $R$ on the metrizable space $X$ for which the equivalence class of each $x \in X$, which is not an integer, consists of the points $x$ and $1/x$, and the equivalence class of each integer consists of that integer alone. Show that $R$ is open and that the graph of $R$ is closed in $X \times X$, but that $X/R$ is not regular.