# Irrational Numbers form G-Delta Set in Reals

## 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.

$\blacksquare$