# A non discrete, torsion-free and $\sigma-$compact, locally compact abelian group?

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?

• A compact example: the $p$-adic integers for any prime $p$. – KCd Jul 20 '13 at 13:47

Yes, $\mathbb{R}$, with addition.
Both $\mathbb{R}$ and $\mathbb{S^1}$ come to mind.
Edit: just kidding. Only $\mathbb{R}$.
• $S^1$ is not torsion-free – Owen Sizemore Jul 20 '13 at 13:24