If $G$ is an additive subgroup of the real numbers $\mathbb{R}$ and $\overline{G}$ is the topological closure of $G$ then either
$\overline{G} = a \cdot \mathbb{Z}$ for some $a \in \mathbb{R}$, or
$\overline{G} = \mathbb{R}$.
If instead $G$ is an additive subgroup of $\mathbb{R}^n$ is there a classification of the the possible values of the topological closure $\overline{G}$ of $G$?
What if $G$ is an additive subgroup of $\mathbb{R}^{\mathbb{N}}$?