From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2436 Path: news.gmane.org!not-for-mail From: Oswald Wyler Newsgroups: gmane.science.mathematics.categories Subject: Re: Uniform spaces Date: Tue, 02 Sep 2003 17:16:50 -0300 Message-ID: References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018662 4159 80.91.229.2 (29 Apr 2009 15:24:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:22 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 2 17:19:48 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Sep 2003 17:19:48 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19uHa6-0001mr-00 for categories-list@mta.ca; Tue, 02 Sep 2003 17:16:50 -0300 In-Reply-To: X-MIME-Autoconverted: from 8bit to quoted-printable by aphrodite.gwi.net id h82IZRWC039976 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 4 Original-Lines: 70 Xref: news.gmane.org gmane.science.mathematics.categories:2436 Archived-At: On Wed, 27 Aug 2003, Tom Leinster wrote: > Date: Wed, 27 Aug 2003 16:51:55 +0200 (CEST) > From: Tom Leinster > To: categories@mta.ca > Subject: categories: Uniform spaces >=20 > Hello, >=20 > Does anyone know of any account of the basic properties of the category= of > uniform spaces? I'm after things like (co)limits, cartesian closure, a= nd > (co)limit-preservation by the forgetful functor to Top. Bourbaki gets = me > some of the way, but his decision not to use categorical language and > the resulting circumlocutions make it a struggle. >=20 > Thanks, > Tom Hi Tom, The category UNIF of uniform spaces, without a separation axiom, is topological over sets, and hence complete and cocomplete, with concrete limits and colimits. UNIF=A0is not cartesian closed. Cook and Fischer, Math. Ann. 173 (1967), 290-306, defined uniform converg= ence structures of a set X as sets \scrF of filters on XxX satisfying five axioms. With the obvious definition of uniform continuity, sets with a uniform convergence structure in this sense form a topological category over sets, but Gazik, Kent and Richardson in Bull.Austral.Math.soc 11 (19= 74), 413-424, showed that this category is not cartesian closed. In LNM 378, 591-637, I replaced the Cook-Fischer axiom that the principal filter generated by the diagonal of XxX is in \scrF by the less demanding axion that the principal filter generated by (x,x), for every x \in X, is in \scrF. This is now part of the accepted definition of uniform convergence spaces. In Bull.Austral.Math.Soc. 15 (1976), 461-465 my student R.S. Lee showed that the category of uniform convergence spaces with this definition is cartesian closed; this is not the cartesian close= d hull of UNIF. For quasitoposes, we must go to semiuniform spaces which have partial morphisms -- relations (m,g) with m an embedding -- represented by one-point extensions. Semiuniform convergence spaces and their uniformly continuous maps form a quasitopos, but not the quasitopos hull of UNIF. This has been determined by Ad=E1mek and Reiterman, The quasitopos hull o= f the category of uniform spaes -- a correction, in the journal Topology and its Applications. For more information and literature, see my book Lecture Notes on Topoi and Quasitopoi. Oswald Wyler