Exercise 11, § 8

Let $X$ be a Hausdorff topological space. Suppose that, for each finite sequence $s$ whose terms are equal to 0 or 1, there is a nonempty subset $A(s)$ of $X$ which satisfies the following conditions: (i) If $s$ is a sequence of $n$ terms (each of which is 0 or 1) and $s', s''$ are the sequences of $n + 1$ terms (each of which is 0 or 1) whose first $n$ terms are the same as those of $s$, then $A(s) = A(s') \cup A(s'')$; and if $s_0$ is the empty sequence, $A(s_0) = X$. (ii) For each infinite sequence $(u_n)_{n \geq 0}$ whose terms are equal to 0 or 1, if $s_n$ denotes the finite sequence $(u_k)_{0 \leq k \leq n}$, then the filter base formed by the $A(s_n)$ converges to a point of $X$. Show that under these conditions there is a surjective continuous mapping of Cantor’s triadic set $K$ onto $X$.