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

No. This was proved by D. van Dantzig in the 1930s:

Van Dantzig, D.: Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen, Compositio Mathematica, Volume 3 (1936), p. 408-426

For a modern presentation of the proof, see e.g. Phillip Wesolek's lecture notes: http://people.math.binghamton.edu/wesolek/tdlc_Polish_groups/tdlcPolish.html

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