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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?

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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 $n\mathbb Z$.

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    $\begingroup$ But $\{0\}$ is a local base for the identity of compact subgroups. $\endgroup$ – Henno Brandsma Nov 1 '18 at 20:31
  • $\begingroup$ You're Right! thanks $\endgroup$ – Can I play with Mathness Nov 1 '18 at 20:35

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