Irrational Number Space is Paracompact

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 paracompact.

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

The result follows from Metric Space is Paracompact.