2
$\begingroup$

Let $G$ be a Hausdorff topological group in which every point has a neighborhood basis of open compact neighborhoods. Let's call this a group of totally disconnected (td)-type.

On the other hand, we have a notion of a locally profinite group, a Hausdorff topological group which has a neighborhood basis of the identity consisting of open compact subgroups. A locally profinite group is obviously of td-tytpe.

Is there an example of a group of td-type which is not locally profinite?

$\endgroup$

2 Answers 2

0
$\begingroup$

I think $\mathbb Z$ would be a counter example : $\mathbb Z$ is discrete thus each point $n$ has a basis of compact open neighborhood given by the compact open set $\{n\}$. Howewer, subgroups of $\mathbb Z$ are of the form $n\mathbb Z$.

$\endgroup$
2
  • 4
    $\begingroup$ But $\{0\}$ is a local base for the identity of compact subgroups. $\endgroup$ Commented Nov 1, 2018 at 20:31
  • $\begingroup$ You're Right! thanks $\endgroup$ Commented Nov 1, 2018 at 20:35
0
$\begingroup$

I think that Van Danztig's Theorem is what you are searching for:

Every totally disconnected locally compact group $G$ contains a compact open subgroup $H$.

See enter link description here for a proof.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .