# Irrational Number Space is Topological Space

## Theorem

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space formed by the irrational numbers $\R \setminus \Q$ under the usual (Euclidean) topology $\tau_d$.

Then $\tau_d$ forms a topology.

## Proof

Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$.

By definition of irrational numbers, $\R \setminus \Q \subseteq \R$.

From Topological Subspace is Topological Space we have that $\struct {\R \setminus \Q, \tau_d}$ is a topology.

$\blacksquare$