Exercise 5, § 6

(a) Prove that every totally disconnected subspace of $\mathbf{R}$ is scattered. (b) Let K be the Cantor triadic set; every $x \in K$ is written in one and only one way $x = \sum_k 2/3^{n_k}$, where $(n_k)_{k \geq 1}$ is a (finite or infinite) strictly increasing sequence of integers $> 0$ (IV, p. 63, Ex. 9); define the number $f(x)$ to be equal to $\sum_k (-1)^{n_k}/2^k$. Let G be the graph of the function $f$ in $K \times \mathbf{R}$; prove that the metrizable space G is totally disconnected, but is not scattered. (Use Ex. 20 of IX, p. 114, applied to a numerical lower semicontinuous function in K, such that the set of $(x, y) \in K \times \mathbf{R}$ such that $y < g(x)$ has an intersection with G which is a set both open and closed.)