$\mathbb{Q}$ is dense in $\mathbb{R}$: every non-empty open interval $(a,b)$ contains a rational number. Equivalently, $\overline{\mathbb{Q}} = \mathbb{R}$. This follows from the Archimedean property of $\mathbb{R}$. $\mathbb{N}$ and $\mathbb{Z}$ are closed and nowhere dense; $\mathbb{R}\setminus\mathbb{Z}$ is open and dense but not equal to $\mathbb{Q}$.