Exercise 14, § 4

Let $X$ be a non-normal Hausdorff space, and let $A, B$ be two disjoint closed subsets of $X$ such that there is no pair of disjoint open sets $U, V$ in $X$ satisfying the relations $A \subset U$ and $B \subset V$. Let $R$ be the equivalence relation on $X$ whose equivalence classes are the set $A$, the set $B$, and the sets $\{x\}$ where $x \in C(A \cup B)$. Show that the graph of $R$ is closed in $X \times X$ and that the relation $R$ is closed, but that the quotient space $X/R$ is not Hausdorff (cf. Chapter I, § 8, no. 3, Proposition 8).