# Every group of totally disconnected type is locally profinite?

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?

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 $$n\mathbb Z$$.
• But $\{0\}$ is a local base for the identity of compact subgroups. – Henno Brandsma Nov 1 '18 at 20:31