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We know that every countable, discrete torsion-free group is $\sigma-$compact. Is there a non discrete, torsion-free, $\sigma-$compact, locally compact abelian group?

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    $\begingroup$ A compact example: the $p$-adic integers for any prime $p$. $\endgroup$ – KCd Jul 20 '13 at 13:47
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Yes, $\mathbb{R}$, with addition.

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Both $\mathbb{R}$ and $\mathbb{S^1}$ come to mind.

Edit: just kidding. Only $\mathbb{R}$.

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  • $\begingroup$ $S^1$ is not torsion-free $\endgroup$ – Owen Sizemore Jul 20 '13 at 13:24
  • $\begingroup$ Haha, yes indeed. $\endgroup$ – MTS Jul 20 '13 at 13:24

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