How to show that in the space X=Q with the usual absolute value metric, the whole space is of first category in X

Question

I know that the union of all singleton sets {q} with q a rational is countable so the issue I'm having is showing that {q} is nowhere dense in X? I know this is true when X=R but having trouble when X=Q

Answer 1

Every open set in the metric space $\Bbb Q$ (with the usual Euclidean absolute value metric) is meagre, i.e., first category in $\Bbb Q$; in fact every subset of $\Bbb Q$ is meagre in $\Bbb Q$, because every subset of $\Bbb Q$ is countable, and $\{x\}$ is a closed, nowhere dense subset of $\Bbb Q$ for each $x\in\Bbb Q$. Hence $\mathbb{Q}$ is of first category in $\mathbb{Q}$.