Irrational Number Space is Non-Meager

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 of the second category.

Proof
From Irrational Number Space is Complete Metric Space, $\left({\R \setminus \Q, d}\right)$ is a complete metric space.

From the Baire Category Theorem, a complete metric space is also a Baire space.

The result follows from Baire Space is Second Category.