Exercise 12, § 8

Deduce from Exer. 11 that: a) If $A$ is a compact subset of $\mathbf{R}$, there exists a continuous mapping of the Cantor triadic set $K$ onto $A$ (take for the $A(s)$ the intersections of $A$ with suitably chosen intervals). b) If moreover $A$ is perfect and totally disconnected, it is homeomorphic to $K$ (same method, proceeding so that if $s_1$ and $s_2$ are two distinct sequences having the same number of terms (equal to 0 or 1), one has $A(s_1) \cap A(s_2) = \varnothing$).