Exercise 11, § 8

Let $E$ be a separated topological space. Suppose that for every finite sequence $s$ whose terms are equal to 0 or to 1, there exists a nonempty subset $A(s)$ of $E$, satisfying the following conditions: 1° If $s$ is a sequence of $n$ terms (equal to 0 or 1), $s', s''$ the sequences of $n + 1$ terms (equal to 0 or 1) whose first $n$ terms are equal to those of $s$, one has $A(s) = A(s') \cup A(s'')$; moreover, if $s_0$ is the empty sequence, $A(s_0) = E$. 2° For every infinite sequence $(u_n)_{n \geq 0}$ whose terms are equal to 0 or to 1, if $s_n$ denotes the finite sequence $(u_k)_{0 \leq k \leq n}$, the filter basis formed by the $A(s_n)$ converges towards a point of $E$. Now prove that under these conditions there exists a continuous surjective mapping from the triadic set of Cantor $K$ into $E$.