# Intersection of Closures of Rationals and Irrationals is Reals

## Theorem

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

Let $\Q$ be the set of rational numbers.

Then:

$\Q^- \cap \paren {\R \setminus \Q}^- = \R$

where:

$\R \setminus \Q$ denotes the set of irrational numbers
$\Q^-$ denotes the closure of $\Q$.

## Proof

$\Q^- = \R$
$\paren {\R \setminus \Q}^- = \R$

The result follows from Intersection is Idempotent.

$\blacksquare$