Nowhere dense sets of product topology

Question

If $X$ = {$0$} $\cup$ {$\frac{1}{n}$: $n\in$ $\Bbb N$}, a metric and topological space, and $Y$ = $X$ x $X$ with the usual $\Bbb R^2$ metric, what are the nowhere dense subsets of $Y$, and the subsets of first category in $Y$?