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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 $\mathbb{R}^{\mathbb{N}}$?

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    $\begingroup$ I have seen the $n$-dimensional case cited as being in Bourbaki's General Topology VII, §1, Theorem 2, but haven't verified. For the other case I do not know, but this makes it seem like the infinite-dimensional case can be quite different. $\endgroup$ Feb 22, 2023 at 6:49
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    $\begingroup$ It's a question about closed subgroups of these various groups. Yes, for $n$ finite: the closed subgroups are indeed the images under $\mathrm{GL}_n(\mathbf{R})$ of the subgroups $\mathbf{R}^k\times\mathbf{Z}^\ell\times\{0\}^m$ for $k+\ell+m=n$. $\endgroup$
    – YCor
    Feb 22, 2023 at 6:51
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    $\begingroup$ I can confirm that this the reference of @Mark is correct (maybe slightly more specifically it's VII, §1.2, Theorem 2). $\endgroup$ Feb 22, 2023 at 10:07

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