Exercise 6, § 4

Show that, in a normal space $X$, every subspace $Y$ of $X$ which is a countable union of closed sets is normal. [Let $A, B$ be two disjoint closed subsets of $Y$, and suppose that $Y$ is the union of an ascending sequence $(Y_n)$ of closed subsets of $X$. Define by induction two sequences $(U_n), (V_n)$ of open subsets of $X$ such that: (i) $U_n \cap Y_n$ and $V_n \cap Y_n$ do not intersect; (ii) $U_n \cap Y_n$ contains $A \cap Y_n$ and all the sets $U_i \cap Y_i$ for $1 \leq i \leq n$; and $V_n \cap Y_n$ contains $B \cap Y_n$ and all $V_i \cap Y_i, 1 \leq i \leq n.$]