Lemma 4

Let $(A_n), (A'_m)$ be two sequences of subsets of a topological space X. Suppose that for each pair $(A_n, A'_m)$ there is a Borel set $B_{nm}$ of X such that $B_{nm} \supset A_n$ and $B_{nm} \cap A'_m = \emptyset$. Then there is a Borel set B of X which contains $\bigcup_n A_n$ and does not meet $\bigcup_m A'_m$.

For the Borel set $B = \bigcup_n \left( \bigcap_m B_{nm} \right)$ satisfies these conditions.