# Irrational Number Space is Completely Normal

## Theorem

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

Then $\struct {\R \setminus \Q, \tau_d}$ is a completely normal space.

## Proof

From Metric Space fulfils all Separation Axioms it follows that $\struct {\R \setminus \Q, \tau_d}$ is a completely normal space.

$\blacksquare$