Exercise 8, § 6

Let X be a Souslin space and let f be a continuous mapping of X into a Hausdorff space Y. Show that if $f(X)$ is uncountable, there exists a subspace A of X homeomorphic to Cantor's triadic set K (Chapter IV, § 2, no. 5) such that $f|A$ is injective. [Reduce to the case where X is a complete metric space: show that there exists a sieve $C = (C_n, p_n)$ such that, for each n, $C_n$ has $2^n$ elements, and that there exists for each n a mapping $\varphi_n$ of $C_n$ into the set of all non-empty closed subsets of diameter $\leq 2^{-n}$ in X, such that (i) $\varphi_{n+1}(c) \subset \varphi_n(p_n(c))$ for all $c \in C_{n+1}$; (ii) whenever c and $c'$ are distinct elements of $C_n$, $f(\varphi_n(c))$ and $f(\varphi_n(c'))$ are disjoint. Use § 5, Exercise 11.] In particular, every uncountable Souslin space contains a subspace homeomorphic to K and therefore has the power of the continuum.