Singletons in $\RR$ are closed, so $\QQ$ is $F_\sigma$

$\QQ$ is $F_\sigma$ because it is a countable union of closed singletons.Suggest a slogan

Every singleton $\theset{x}\subseteq\RR$ is closed, and $\QQ=\bigcup_{q\in\QQ}\theset{q}$ is a countable union of closed sets, hence an $F_\sigma$ set in $\RR$.