Rational Numbers form F-Sigma Set in Reals

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Theorem

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

Let $\left({\R, \tau}\right)$ be the real number line under 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 Space, $\left\{{\alpha}\right\}$ is a closed set of $\R$.

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

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

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

$\blacksquare$


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