Might be interesting to note that Cauchy and filters with convergence etc can be described now in more generality involving underlying (partially ordered) monads. See e.g. http://books.google.com/books?id=EoJGeBZVOaQC&pg=PA65&lpg=PA65&dq=%22partially+ordered+monad%22+eklund&source=web&ots=Jg6-Pbj-z5&sig=lFHkMQ-_VjvQMxHMfTryuHprKPQ#PPA102,M1 http://sevein.matap.uma.es/~aciego/mat-es/patrik.pdf Cheers, Patrik On Mon, 11 Feb 2008, Toby Bartels wrote:
Jeff Egger wrote in part:
one could define a uniform locale as a locale equipped with a designated set of Cauchy filters
This can be done, although you get a more general notion: a _Cauchy_space_ (or locale, but spaces have a bigger literature). Note that a uniform space is a Cauchy space with extra ~structure~; there is no way, in a mere Cauchy space, to compare sizes of neighbourhoods of different points.
For the basic definition, you could do worse than the English Wikipedia: < http://en.wikipedia.org/wiki/Cauchy_space >. I had some references (monographs) that I liked too, but I can't find them now; perhaps I can find them tomorrow.
--Toby