Exercise 28, § 8

Let $X$ be a separated non-regular space, $F$ a closed subset of $X$, $a \notin F$ a point of $X$ such that every neighbourhood of $a$ meets every neighbourhood of $F$. Let R be the equivalence relation obtained by identifying with one another all the points of $F$. Now prove that R is closed and that its graph is closed in $X \times X$, but that $X/R$ is not separated.