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>



the meaning of "Cauchy filter" for an arbitrary topological group
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](http://en.wikipedia.org/wiki/Uniform_space#Completeness). 


