Exercise 3, § 1

Let $A$ and $B$ be two countable dense subsets of $\mathbf{R}^2$. Show that there exists a homeomorphism of $\mathbf{R}^2$ onto itself which maps $A$ onto $B$. (Show first that, by means of a rotation, we can reduce to the case where the projections $pr_1$ and $pr_2$ are injective mappings of $A$ and $B$ into $\mathbf{R}$. Then define, by a suitable inductive process, a bijection of $A$ onto $B$ which determines a monotone mapping of $pr_1 A$ onto $pr_1 B$ and a monotone mapping of $pr_2 A$ onto $pr_2 B$. Finally, using Exercise 11, Chapter IV, § 2, show that this mapping is a homeomorphism which can be extended to a homeomorphism of $\mathbf{R}^2$ onto itself.) Deduce that, if $C$ is a subset of $\mathbf{R}^2$ whose complement is dense, $C$ is homeomorphic to a subset of the complement of $\mathbf{Q}^2$ in $\mathbf{R}^2$. Generalize these results to $\mathbf{R}^n, n > 2$.