Exercise 12, § 2

Let $X$ be a compact metrizable space.

a) Show that there is a continuous mapping of Cantor's triadic set $K$ (Chapter IV, § 2, no. 5) onto $X$ (cf. Chapter IV, § 8, Exercise 11).

b) If in addition $X$ is totally disconnected and has no isolated points, then $X$ is homeomorphic to $K$. [Argue as in Chapter IV, § 8, Exercise 12, using the fact that every neighbourhood of a point $x \in X$ contains a neighbourhood of $x$ which is both open and closed (Chapter II, § 4, no. 4, Corollary to Proposition 6)].