Constructing a continuous simple approximation of the identity map for a totally disconnected compact metric space

Question

In Petersen's Ergodic Theory, when talking about totally disonnected spaces (Section 4.4), he says that if $D$ is a compact, totally disconnected subspace of $\mathbb{R}$, then for every $\delta > 0$, there exists a continuous simple function $\phi : D \to D$ taking values $\{ d_1, \ldots, d_n \} \subseteq D$ such that $\phi(d_j) = d_j$ and $\operatorname{diam} \phi^{-1} \{d_j\} < \delta$ for all $j$. How does he conclude this? If I'm not mistaken, this is equivalent to saying that there's a clopen neighborhood basis for each point in $D$, but I don't know how to prove this. I understand that this latter property is called being zero-dimensional, and that it's in general a stronger property than being totally disconnected, but I'm not sure how to show that in this case, the weaker claim implies the stronger.

Answer 1

According to https://en.wikipedia.org/wiki/Zero-dimensional_space

[See the heading "Properties of spaces with small inductive dimension zero"] a locally compact space is totally disconnected iff it is zero dimensional. A reference for the proof is also given there.