# Rational Numbers form F-Sigma Set in Reals

## Theorem

Let $\Q$ be the set of rational numbers.

Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology.

Then $\Q$ is a $F_\sigma$ set in $\R$.

## Proof

Let $\alpha \in \Q$ be a rational number.

From Real Number is Closed in Real Number Line, $\set \alpha$ is a closed set of $\R$.

From Rational Numbers are Countably Infinite, $\displaystyle \bigcup_{\alpha \mathop \in \Q} \set \alpha$ is a countable union.

Thus $\Q = \displaystyle \bigcup_{\alpha \mathop \in \Q} \set \alpha$ is a countable union of closed sets of $\R$.

Hence the result by definition of $F_\sigma$ set.

$\blacksquare$