From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5982 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: non-Hausdoff topology Date: Fri, 09 Jul 2010 15:10:45 +0100 Message-ID: <4C372DE5.10906@cs.bham.ac.uk> References: Reply-To: Steve Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1278765749 3460 80.91.229.12 (10 Jul 2010 12:42:29 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 10 Jul 2010 12:42:29 +0000 (UTC) Cc: categories list To: Vaughan Pratt Original-X-From: categories@mta.ca Sat Jul 10 14:42:27 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 1OXZNz-00012Q-6Y for gsmc-categories@m.gmane.org; Sat, 10 Jul 2010 14:42:27 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OXYu2-0003ka-7e for categories-list@mta.ca; Sat, 10 Jul 2010 09:11:30 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5982 Archived-At: Dear Vaughan, The Zariski topology on the prime spectrum is normally not T1. In fact Hochster showed that any spectral space (the ones that correspond to ordered Stone spaces in the Priestly duality) is homeomorphic to the prime spectrum of some ring, with its Zariski topology. In particular, this holds for Scott domains and some other classes of spaces commonly arising in denotational semantics - though it would be eccentric to treat them as spectra of rings. Referring to the Wikipedia page on Zariski topology, the "classical" definition gives a T1 topology, but the "modern" definition adds extra "generic" points corresponding to non-maximal prime ideals and they create a non-discrete specialization order, T0 but not T1. Regards, Steve. Vaughan Pratt wrote: > > On 7/7/2010 10:28 AM, Michael Barr wrote: >> Not just in CS, but also central to algebraic geometry: the Zariski >> topology is almost never hausdorff. But when topology is taught to >> undergraduates, it is usually for the purposes of analysis and I don't >> know if we could this point across. > > My understanding of Paul's complaint was with the passage not so much > from T2 to T0 but from T1 to T0, needed for the Scott topology. The > Zariski topology is always T1. > > Vaughan Pratt > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]