Exercise 28, § 8

Let $X$ be a Hausdorff space which is not regular, let $F$ be a closed subset of $X$ and let $a \notin F$ be a point of $X$ such that every neighbourhood of $a$ meets every neighbourhood of $F$. Let $R$ be the equivalence relation on $X$ obtained by identifying all the points of $F$. Show that $R$ is closed and that the graph of $R$ is closed in $X \times X$, but that $R$ is not Hausdorff.