Exercise 19, § 4

(a) Let $X$ be a normal space, $R$ a closed equivalence relation in $X$. Now prove that the quotient space $X/R$ is normal (use prop. 10 of I, p. 35).

(b) Let $X$ be a locally compact space countable at infinity, $R$ an equivalence relation in $X$ whose graph is closed in $X \times X$. Now prove that $X/R$ is normal (cf. I, p. 113, exerc. 19).

(c) Let $X$ be the complement, in $\mathbf{R}$, of the set of points of the form $1/n$, where $n$ runs through the set of integers different from 0 and from $\pm 1$. We consider, in the metrizable space $X$, the equivalence relation $R$ for which the equivalence class of every non-integer point $x \in X$ is formed by the points $x$ and $1/x$, and the equivalence class of each integer is reduced to that integer. Now prove that $R$ is open and has a closed graph in $X \times X$, but that $X/R$ is not regular.