Exercise 10, § 8

Let $X$ be the subspace of $\mathbf{R}$ which is the complement of the set of points $1/n$, where $n$ is a rational integer other than $0, 1$ or $-1$. Let $S$ be the equivalence relation on $X$ whose equivalence classes are (i) the set $\{ 0 \}$; (ii) the set of all non-zero integers; (iii) all sets of the form $\{ x, 1/x \}$ where $x \in X$ and $0 < |x| < 1$.

a) Show that the relation $S$ on $X$ is neither open nor closed.

b) Show that the graph of $S$ is closed in $X \times X$, but that the quotient space $X/S$ is not Hausdorff. \*