Exercise 6, § 5

Let $S$ be the equivalence relation on the rational line $\mathbf{Q}$ which is obtained by identifying all the points of $\mathbf{Z}$. Show that $S$ is closed, and that if $U$ is the relation of equality on $\mathbf{Q}$, then the canonical bijection of $(\mathbf{Q} \times \mathbf{Q})/(U \times S)$ onto $\mathbf{Q} \times (\mathbf{Q}/S)$ is not a homeomorphism.