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As we know the symbol $n$ determines the dimension of $\mathbb{R}^n$ in the category of vector spaces. Also, in the category of manifolds, $n$ determines the dimension of $n$-manifold $\mathbb{R}^n$.

Now, my question is that:

Is there any algebraic property related to the additive group $(\mathbb{R}^n ,+)$ which is determined by $n$?

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This answer proves that as additive groups $\mathbb{R}^n \cong \mathbb{R}^m$ for all $n,m$.

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    $\begingroup$ As $K$-vector spaces even, for any countable subfield $K$ of $\mathbb{R}$ $\endgroup$ – Maxime Ramzi Aug 29 '18 at 8:57
  • $\begingroup$ @IvanDiLiberti Thank you very much. $\endgroup$ – M.Ramana Aug 29 '18 at 11:16
  • $\begingroup$ @Max This is a good point for me. Thank you so much. $\endgroup$ – M.Ramana Aug 29 '18 at 11:19

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