F-sigma set

Remarks

A set is $F_\sigma$ if and only if its complement is a G-delta set, by De Morgan's laws.

Example: $\Q = \bigcup_{q \in \Q} \set{q}$ in $\R$. Each $\set{q}$ is closed and $\Q$ is countable, so $\Q$ is $F_\sigma$.

Example: $(0, 1) = \bigcup_{n \ge 3} [1/n, 1 - 1/n]$, so this open interval is $F_\sigma$. In a metric space, every open set is $F_\sigma$.

Non-example: the set $\R \setminus \Q$ of irrational numbers is not $F_\sigma$ in $\R$. Otherwise its complement $\Q$ would be a $G_\delta$ set, and it is not.