Exercise 3, § 6

a) Show that every totally disconnected subspace of $\mathbf{R}$ is zero-dimensional.

b) Let $K$ be Cantor’s triadic set; every $x \in K$ can be written uniquely in the form $x = \sum_{k} 2/3^{n_k}$, where $(n_k)_{k \geq 1}$ is a strictly increasing (finite or infinite) sequence of integers $> 0$ (Chapter IV, § 8, Exercise 9); and define $f(x)$ to be the number $\sum_{k} (-1)^{n_k}/2^k$. Let $G$ be the graph of the function $f$ in $K \times \mathbf{R}$. Show that the metrizable space $G$ is totally disconnected but is not zero-dimensional. [Note that the intersection of $\overline{G}$ and the line $\{0\} \times \mathbf{R}$ contains an open interval containing the point $(0, 0)$.]