A topological space $X$ is a normal space if, given any disjoint closed sets $E$ and $F$, there are open neighbourhoods $U$ of $E$ and $V$ of $F$ that are also disjoint. (Or more intuitively, this condition says that $E$ and $F$ can be separated by neighbourhoods.)
And an $F_{\sigma}$-set is a countable union of closed sets.
So I should be able to show that the $F_{\sigma}$-set has the necessary conditions for a $T_4$ space? But how could I for instance select two disjoint closed sets from $F_{\sigma}$?
Best Answer
$\newcommand{\cl}{\operatorname{cl}}$We can actually prove more. Let $X$ be a $T_4$-space, and let $A$ be an $F_\sigma$-set in $X$; if $H$ and $K$ are disjoint, relatively closed subsets of $A$, then there are disjoint open sets $U$ and $V$ in $X$ (not just in $A$) such that $H\subseteq U$ and $K\subseteq V$.
One proof of this is very similar to the usual ‘climbing a chimney’ proof that a regular, Lindelöf space is normal.