From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5978 Path: news.gmane.org!not-for-mail From: Steven Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: non-Hausdoff topology Date: Thu, 08 Jul 2010 17:45:28 +0100 Message-ID: References: Reply-To: Steven Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1278684004 16413 80.91.229.12 (9 Jul 2010 14:00:04 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 9 Jul 2010 14:00:04 +0000 (UTC) Cc: To: Vaughan Pratt Original-X-From: categories@mta.ca Fri Jul 09 16:00:03 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OXE7W-0002At-S4 for gsmc-categories@m.gmane.org; Fri, 09 Jul 2010 16:00:03 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OXDe6-0004Uk-Fy for categories-list@mta.ca; Fri, 09 Jul 2010 10:29:38 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5978 Archived-At: Dear Vaughan, The problem with these representations (Nachbin, Priestley, ...) is that they are not completely functorial. The natural morphisms (order-preservi= ng continuous maps) between the Nachbin or Priestley spaces correspond not t= o arbitrary continuous maps between the non-Hausdorff spaces, but to the perfect ones. Scott's insight was that computability implied a form of continuity, henc= e the role of the Scott topology in domains. If you restrict to perfect map= s then you lose some computable maps. To put it rudely, the order (and the fact that continuity implies monotonicity and preservation of directed joins) is already naturally present in topological spaces, and the _un_natural thing to do is (1) deliberately exclude it, then (2) put it back artificially, and (3) ignor= e the fact that you don't quite get the same thing. The question of whether one likes the traditional notion of topological space is really focusing on the wrong things - the objects instead of the morphisms. (Actually, the traditional notion is not that attractive.) The big question is how one explains continuity, and topological spaces were set up to give an abstract definition of it. Scott's insights have elucidated continuity for us, and at the same time validated the non-Hausdorff notion of topological space (as I tried to explain in my book). And let me take this much further: Grothendieck's insights into continuit= y have shown that topological spaces don't go far enough. For example, he showed that it is good to extend one's ideas of continuity so that a continuous map to the "space of sets" (which doesn't exist as a topologic= al space) is a sheaf, and a continuous map from that space is a functor preserving filtered colimits. His technique for topologization - specify the category of sheaves - goes far beyond Hausdorff spaces and brings in specialization orders that are not only non-discrete but even not orders (they are the morphism structures in categories of points). Again, the functoriality and preservation of filtered colimits is a natural and intrinsic part of this. (And we also know that when we follow Grothendieck's relativization programme along these lines then we end up using point-free spaces rather than the ordinary point-set spaces.) To conclude, I don't believe that struggles to keep topology Hausdorff ar= e compatible with the deep insights of Scott, Grothendieck and others who have given us important new clues to the nature of continuity. Best wishes, Steve. On Wed, 07 Jul 2010 06:35:07 -0700, Vaughan Pratt wrote: > I thought the point of the Lawson topology was to show the opposite: > that the benefits of the Scott topology could be had without having to > broaden topology beyond Hausdorff. >=20 > But if one were to so broaden it, wouldn't it be more natural to do so = a > la Nachbin and Priestly, with topologized posets? >=20 > But if you really like the traditional notion of a topological space in > all its generality, why insist on the closure conditions on open sets > when we know that dropping them gives a category with reasonable > properties, namely extensional Chu(Set,2), further improved to a very > nice category by dropping extensionality, and generalizable to > Chu(Set,K) and yet further to Chu(V,k)? >=20 > Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]