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

Question

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 — KCd, Jul 20 '13 at 13:47

score 1 · Answer 1 · answered Jul 20 '13 at 13:23

1

Yes, $\mathbb{R}$, with addition.

answered Jul 20 '13 at 13:23

Owen Sizemore

6,440

score 0 · Answer 2 · answered Jul 20 '13 at 13:24

0

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

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

answered Jul 20 '13 at 13:24

MTS

386

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

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

2 Answers2