Exercise 10, § 4

Prove now that, in a normal space X, every subspace Y which is a countable union of closed sets, is normal. (Let A, B be two closed subsets of Y, with no common point, and suppose that Y is the union of an increasing sequence $(F_n)$ of closed subsets of X. Define by induction two sequences $(U_n), (V_n)$ of open subsets of X, such that: $1^\circ \overline{U_n} \cap F_n$ and $\overline{V_n} \cap F_n$ are disjoint; $2^\circ U_n \cap F_n$ contains $A \cap F_n$ and all the sets $\overline{U_i} \cap \overline{F_i}$ for $1 \leq i \leq n$, and $V_n \cap F_n$ contains $B \cap F_n$ and all the sets $\overline{V_i} \cap \overline{F_i}$ for $1 \leq i < n$.