# Irrational Numbers form G-Delta Set in Reals

## 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

$\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.

$\blacksquare$