Exercise 12, § 1

a) Let $f$ be a mapping of $\mathbf{R}$ into $\mathbf{R}$, and let $G \subset \mathbf{R}^2$ be the graph of $f$. Suppose that $G$ is dense in $\mathbf{R}^2$ and meets every perfect set in $\mathbf{R}^2$ which is not contained in any countable union of lines $\{x_n\} \times \mathbf{R}$. Show that $G$ is connected. (If this were false, show that there would exist two non-empty disjoint open sets $A, B$ in $\mathbf{R}^2$ such that $G$ was the union of $G \cap A$ and $G \cap B$, and there would exist at least one point of $A$ and one point of $B$ with the same abscissa, by using the fact that $\mathbf{R}$ is connected; then use Exercise 11.)

b) Deduce from a) that there is a linear mapping $f$ of $\mathbf{R}$ into $\mathbf{R}$ whose graph $G$ is dense in $\mathbf{R}^2$ and connected. [Using a) and Exercise 11, define by transfinite induction the values of $f$ at the points of a Hamel base of $\mathbf{R}$, using the same method as in Set Theory, Chapter III, § 6, Exercise 24.] Deduce that the subgroup $G$ of $\mathbf{R}^2$ satisfies conditions (R_I) and (R_{II}) of Chapter V, § 3, Exercise 4, but is not locally compact, and therefore has a topology strictly finer than the usual topology of $\mathbf{R}$; and that $G$ is not locally connected.