Uniformity via Cauchy filters?
Dear categorists, One of my current research projects has drifted into the realm of uniform locales, which is not exactly my forte. As I went to re-familiarise myself with the assorted definitions, I found myself wondering whether one could define a uniform locale as a locale equipped with a designated set of Cauchy filters, instead of uniform covers, or entourages. Does anyone know whether this idea has already been pursued, either for locales or for spaces? Cheers, Jeff. Looking for the perfect gift? Give the gift of Flickr! http://www.flickr.com/gift/
Hi Jeff, I don't know about locales, but I worked on issues surrounding taking a uniform space, and forgetting the uniformity but keeping the set of cauchy filters. This turns out to be a nontrivial amount of forgetting. I liked the result because the resulting category, which I call the category of completable spaces, can be used as a base category and, for example, if you have a commutative ring, you can look at the compatible completabilities on it (making it a completable space with the discrete completability) and they generalize the valuations on a field. Bill Rowan On Sun, 10 Feb 2008, Jeff Egger wrote:
Dear categorists,
One of my current research projects has drifted into the realm of uniform locales, which is not exactly my forte. As I went to re-familiarise myself with the assorted definitions, I found myself wondering whether one could define a uniform locale as a locale equipped with a designated set of Cauchy filters, instead of uniform covers, or entourages. Does anyone know whether this idea has already been pursued, either for locales or for spaces?
Cheers, Jeff.
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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
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
I wrote in part:
I had some references (monographs) that I liked too, but I can't find them now; perhaps I can find them tomorrow.
I've (re-)found this one: Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989. This has many pretty diagrams of various categories of spaces, all included in one another by bireflections and the like. The whole subject seems messy, as point-set topology can be, (there are too many categories around to keep track of), and probably needs to be cleaned up somewhat before outsiders (including me) can get a grasp on it; in particular, perhaps locales would be a better approach. OTOH, my reference is nearly 20 years old, and perhaps the situation has improved. Indeed, perhaps this is what Paul Ekland's references are doing; I haven't looked at them carefully yet. --Toby
participants (4)
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Bill Rowan -
Jeff Egger -
Patrik Eklund -
Toby Bartels