Irrational Number Space is Completely Normal

Theorem
Let $\left({\R \setminus \Q, \tau}\right)$ be the set of all irrational number under the usual (Euclidean) topology.

Then $\left({\R \setminus \Q, \tau}\right)$ is a completely normal space.

Proof
From Euclidean Space is Complete Metric Space, a Euclidean space is a metric space.

From Metric Space fulfils all Separation Axioms it follows that $\left({\R \setminus \Q, \tau}\right)$ is a completely normal space.