Irrational Numbers form G-Delta Set in Reals

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Theorem

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

Let $\left({\R, \tau}\right)$ be the real number line under the usual (Euclidean) topology.


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


Proof

\(\displaystyle \Q\) \(=\) \(\displaystyle \bigcup_{\alpha \mathop \in \Q} \left\{ {\alpha}\right\}\) Rational Numbers form F-Sigma Set in Reals
\(\displaystyle \implies \ \ \) \(\displaystyle \R \setminus \Q\) \(=\) \(\displaystyle \R \setminus \bigcup_{\alpha \mathop \in \Q} \left\{ {\alpha}\right\}\)
\(\displaystyle \) \(=\) \(\displaystyle \bigcap_{\alpha \mathop \in \Q} \left({\R \setminus \left\{ {\alpha}\right\} }\right)\) De Morgan's Laws: Difference with Union

The result follows from Rational Numbers are Countably Infinite.

$\blacksquare$


Sources