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Let $G$ be a topological group which is totally disconnected. Then one point sets in $G$ are closed, and hence $G$ is Hausdorff.

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 totally disconnected.

Is there an example of a totally disconnected topological group which is not locally profinite?

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    $\begingroup$ $\mathbb Q{}{}$ $\endgroup$ – Wojowu Nov 1 '18 at 17:26
  • $\begingroup$ I did not formulate my question the way I wanted to, I wanted totally disconnected + locally compact $\endgroup$ – D_S Nov 1 '18 at 20:20
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$\mathbb{Q}$ is an example, as are the irrationals (in the guise of the group) $\mathbb{Z}^\omega$ in the product topology. Both spaces are not locally compact at any point of the space.

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  • $\begingroup$ Okay, I realized I did not ask the right question. I want the disconnected topological group to also be locally compact. $\endgroup$ – D_S Nov 1 '18 at 20:08
  • $\begingroup$ math.stackexchange.com/questions/2980909/… $\endgroup$ – D_S Nov 1 '18 at 20:14

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