MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
First, since you're only speaking of neighborhoods of identity, you may as well assume G=K is compact. Anyway, a Hausdorff topological group is metrizable if and only if it has a countable basis of neighborhoods of the identity (that is the Birkhoff-Kakutani theorem)- so the criterion you're looking for is simply the metrisability of the group. – Julien Melleray May 26 '12 at 9:42
it seems to me that a product over an uncountable set, something like $\prod_{x\in \mathbb R} \mathbb F_p$ will not have a countable basis. – vytas May 26 '12 at 9:58
the terminology "locally profinite" is in contradiction with the mainstream use of "locally" in topological group theory. Its natural meaning would be: every compact subset is contained in a compact open subgroup. Many people deal with totally disconnected LC-groups (LC= locally compact) and they're only referred as "locally profinite" by a few people, mainly around Langlands's theory. – YCor May 26 '12 at 13:38
Number of mathscinet/google references: "locally profinite group": 11/4220 "totally disconnected locally compact group": 60/22700; "locally compact totally disconnected group" 23/16300. Wikipedia contains some nontrivial general nontrivial facts (Willis' theory) about totally disconnected LC-groups, inside the page "totally disconnected groups", but also redirects to a page "locally profinite groups" which essentially contains nothing. I hope the latter page will be renamed but I'm not technically qualified to do this. – YCor May 26 '12 at 13:54
up vote 0 down vote accepted

Check out Casselman's notes. They are quite enlightening, and just came up yesterday for another question.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.