Irrational Number Space is Paracompact

Theorem
Let $\left({\R \setminus \Q, \tau_d}\right)$ be the irrational number space under the Euclidean topology $\tau_d$.

Then $\left({\R \setminus \Q, \tau_d}\right)$ is paracompact.

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

The result follows from Metric Space is Paracompact.