Closed Set - Closed Set = $F_{\sigma}$ set?

Question

Let $A$ and $B$ be closed sets in $\mathbb{R}$. Is $A\setminus B$ an $F_{\sigma}$ set?

Answer 1

The set difference $A-B$ is the same as $A \cap B'$ where $B'$ means the complement of $B$ and so is open. So show that an open set is also an $F_\sigma$ set, which is to say a countable union of closed sets. To do this, use that an open set is a countable union of pairwise disjoint open intervals, and each of those is the union of a countable increasing family of closed intervals.