Consider a locally profinite group $G$, i.e. a locally compact, totally disconnected topological group. Suppose it admits an open maximal compact subgroup named $K$.
It is known that $G$ admits as a neighborhood basis of the identity element a collection $\{ K\_i \}$ of open compact subgroups, but what can we say about the countability of this family? 


In the basic examples I know, $GL\_1(\mathbb{Q}\_p)$ and $GL\_2(\mathbb{Q}\_p)$, the family ${K\_i}$ is in fact countable, is it always true?
Any reference is greatly appreciated, thanks.



Neighborhood basis of the identity in a locally profinite group
Check out [Casselman's notes.](http://www.math.ubc.ca/~cass/research/pdf/Profinite.pdf) They are quite enlightening, and just came up yesterday for [another question.](https://mathoverflow.net/questions/97971/haar-measure-for-profinite-groups-reference-needed)


