# Irrational Numbers form G-Delta Set in Reals

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## Theorem

Let $\R \setminus \Q$ denote the set of irrational numbers.

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

Then $\R \setminus \Q$ forms a $G_\delta$ set in $\R$.

## Proof

\(\ds \Q\) | \(=\) | \(\ds \bigcup_{\alpha \mathop \in \Q} \set \alpha\) | Rational Numbers form F-Sigma Set in Reals |
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\(\ds \leadsto \ \ \) | \(\ds \R \setminus \Q\) | \(=\) | \(\ds \R \setminus \bigcup_{\alpha \mathop \in \Q} \set \alpha\) | |||||||||||||

\(\ds \) | \(=\) | \(\ds \bigcap_{\alpha \mathop \in \Q} \paren {\R \setminus \set \alpha}\) | De Morgan's Laws: Difference with Union |

The result follows from Rational Numbers are Countably Infinite.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $31$. The Irrational Numbers: $2$