Exercise 14, § 5

Let $X$ be the subspace of $\mathbf{R}^2$ consisting of the points $(r, 0)$, where $r$ runs through the set $\mathbf{Q}$ of rational numbers, and the points $(k/n, 1/n)$, where $n$ runs through the set of integers $\geqslant 1$ and $k$ runs through the set of all rational integers. Show that $X$ is a Baire space but is not totally non-meagre.