Irrational Number Space is Topological Space

Theorem
The set $\R \setminus \Q$ of irrational numbers under the Euclidean topology $\tau$ forms a topology.

Proof
The real Euclidean space $\left({\R, \tau}\right)$ is a topology.

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

From Topological Subspace is Topological Space we have that $\left({\R \setminus \Q, \tau}\right)$ is a topology.