# Rational Numbers form F-Sigma Set in Reals

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## 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.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $30$. The Rational Numbers: $2$