Exercise 18, § 4

Let $X$ be a non-normal separated space, $A$ and $B$ two closed sets without a common point in $X$, such that there exists no pair of open sets $U, V$ without a common point, satisfying the relations $A \subset U, B \subset V$. Let $R$ be the equivalence relation in $X$ whose equivalence classes are the set $A$, the set $B$ and the sets $\{x\}$, where $x$ runs through $C(A \cup B)$. Prove that the graph of $R$ in $X \times X$ is closed and that the relation $R$ is closed, but that the quotient space $X/R$ is not separated (cf. I, p. 155, prop. 8).