# Closure of Intersection of Rationals and Irrationals is Empty Set

## Theorem

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

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

Then:

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

where:

$\R \setminus \Q$ denotes the set of irrational numbers
$\paren {\Q \cap \paren {\R \setminus \Q} }^-$ denotes the closure of $\Q \cap \paren {\R \setminus \Q}$.

## Proof

$\Q \cap \paren {\R \setminus \Q} = \O$

By Empty Set is Closed in Topological Space, $\O$ is closed in $\R$.

$\O^- = \O$

Hence the result.

$\blacksquare$