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, $\ds \bigcup_{\alpha \mathop \in \Q} \set \alpha$ is a countable union.
Thus $\Q = \ds \bigcup_{\alpha \mathop \in \Q} \set \alpha$ 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: $2$