# Is every totally disconnected topological group locally profinite?

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?

• $\mathbb Q{}{}$ – Wojowu Nov 1 '18 at 17:26
• I did not formulate my question the way I wanted to, I wanted totally disconnected + locally compact – D_S Nov 1 '18 at 20:20

$$\mathbb{Q}$$ 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.