Rational Number Space is Meager

Theorem
Let $\Q$ be the set of all rational numbers.

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

Then $\left({\Q, d}\right)$ is of the first category.

Proof
From Rational Numbers are Countably Infinite, $\Q$ is the union of singleton subsets.

From Singleton Sets are Nowhere Dense in Rational Space, each of those singleton subsets is nowhere dense in $\Q$.

The result follows from definition of first category.