Theorem 5

Let X be a metrizable space and let Y be a relatively compact Souslin subspace of X. Then Y is capacitable with respect to every capacity f on X.

We have seen (no. 2, Proposition 9) that there exists a compact space K, a decreasing sequence $(A_n)$ of subsets of K, each of which is a countable union of compact sets, and a continuous mapping $\varphi : K \to X$ such that $Y = \varphi \left( \bigcap_n A_n \right)$. By Proposition 14, $\bigcap_n A_n$ is capacitable with respect to every capacity on K. Theorem 5 is therefore a consequence of the following proposition: