2
$\begingroup$

I was reading a definition of pro-Lie group and it spoke of a "Cauchy filter" on an arbitrary topological group even though there was no mention of a metric. Is there some kind of standard meaning for "Cauchy filter" in the context of an arbitrary topological group, or perhaps there was some kind of assumption being made that there is a metric which is compatible with the topology?

The source I was reading is here:

http://www.matha.rwth-aachen.de/publikationen/pumpluen-festschrift/Hofmann_Morris.pdf

$\endgroup$
1

1 Answer 1

6
$\begingroup$

There is a notion of uniform space, developed by Weil in the 40's or 50's, that generalizes the notion of metric space. Once you realize that topological groups carry standard uniform space structures (actually two: a right uniformity and left uniformity, which coincide if the group is abelian and which at any rate yield the same underlying topology), then you can read about Cauchy filters and completeness starting here.

$\endgroup$
5
  • 2
    $\begingroup$ A small addendum about agreement between the left and right uniformities on a topological group. These uniformities agree not only for abelian groups but also for compact groups. In fact, in the compact case, you don't need the group structure; any compact Hausdorff topology is induced by a unique uniformity. $\endgroup$ Oct 13, 2014 at 13:27
  • $\begingroup$ So, in fact, "Cauchy filter" is not standard in a general topological group. Instead, "left Cauchy" and "right Cauchy" filters are. $\endgroup$ Oct 13, 2014 at 14:36
  • 2
    $\begingroup$ The topological groups where the left uniformities and the right uniformities coincide are known as SIN (Small Invariant Neighborhood) groups. These are precisely the groups $G$ such that the identity $e$ has a basis consisting of sets $U$ such that $xUx^{-1}=U$ for all $x\in G$. $\endgroup$ Oct 13, 2014 at 14:52
  • $\begingroup$ I've also seen the term "balanced group"; I don't know how standard that is. $\endgroup$ Oct 13, 2014 at 16:17
  • $\begingroup$ @GeraldEdgar Yes. It might be worth adding that a $T_0$ (therefore Hausdorff) group admits a topological group completion iff the right and left Cauchy filters coincide. $\endgroup$
    – Todd Trimble
    Oct 13, 2014 at 17:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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