Irrational Number Space is Non-Meager

Theorem
Let $\mathbb I = \R \setminus \Q$ be the set of all irrational numbers.

Let $d: \mathbb I \times \mathbb I \to \R$ be the Euclidean metric on $\mathbb I$.

Then $\left({\mathbb I, d}\right)$ is of the second category.

Proof
From Irrational Number Space is Complete Metric Space, $\left({\mathbb I, d}\right)$ 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.