Rational Numbers are F-Sigma Set in Real Line

Theorem
Let $\left({\R, \tau}\right)$ be the real number line considered asa topological space with the usual (Euclidean) topology.

Then:
 * $\Q$ is an $F_\sigma$ set in $\left({\R, \tau}\right)$.

Proof
Define the set of subsets of $\R$ as:
 * $\mathcal F := \left\{{\left\{{x}\right\}: x \in \Q}\right\}$

By Closed Real Interval is Closed Set:
 * $\forall x \in \Q: \left[{x \,.\,.\, x}\right] = \left\{{x}\right\}$ is closed (in topological sense)

Then:
 * $\forall A \in \mathcal F: A$ is closed

By Cardinality of Set of Singletons:
 * $\left\vert{\mathcal F}\right\vert = \left\vert{\Q}\right\vert$

where $\left\vert{\mathcal F}\right\vert$ denotes the cardinality of $\mathcal F$.

By Rational Numbers are Countably Infinite:
 * $\Q$ is countable.

Therefore by Set is Countable if Cardinality equals Cardinality of Countable Set:
 * $\mathcal F$ is countable.

By Union of Set of Singletons:
 * $\bigcup \mathcal F = \Q$

Thus, by definition, $\Q$ is an $F_\sigma$ set.