Irrational Numbers form G-Delta Set in Reals

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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 $\R$.


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

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

The result follows from Rational Numbers are Countably Infinite.