Irrational Number Space is Non-Meager

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 non-meager.

Proof
From Irrational Number Space is Complete Metric Space, $\struct {\R \setminus \Q, d}$ 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 Non-Meager.