From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6437 Path: news.gmane.org!not-for-mail From: Dana Scott Newsgroups: gmane.science.mathematics.categories Subject: Re; Does this topology have a name? Date: Sun, 2 Jan 2011 10:30:32 -0800 Message-ID: References: <7B51C5A3-329E-4BA4-A0E5-35A4F7E309E0@cs.cmu.edu> Reply-To: Dana Scott NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1082) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1294013524 4090 80.91.229.12 (3 Jan 2011 00:12:04 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 3 Jan 2011 00:12:04 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Mon Jan 03 01:11:58 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PZY1l-0007XH-9T for gsmc-categories@m.gmane.org; Mon, 03 Jan 2011 01:11:57 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:51181) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PZY1c-0001nC-3W; Sun, 02 Jan 2011 20:11:48 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PZY1O-0000yI-G5 for categories-list@mlist.mta.ca; Sun, 02 Jan 2011 20:11:34 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6437 Archived-At: I wrote to Barr as follows: > From: Dana Scott > Date: January 1, 2011 2:57:30 PM PST > To: Michael Barr > Subject: Re: categories: Does this topology have a name? > > > On Jan 1, 2011, at 2:15 PM, Michael Barr wrote: > >> Yes, the fact that when these sets are taken as clopens >> gives a Stone space is easy. But I want to know what to >> call the weaker topology in which you take these sets as >> a basis of opens. > > Ah, I had thought you meant the clopen case. The weaker > topology is (unfortunately) called the Scott topology, > which can be given to any algebraic lattice. The > congruences form an algebraic lattice inasmuch as they > are closed under arbitrary intersections and directed > unions. (Yes?) Details are in the book: Continuous > Lattices and Domains by Gierz/Hofmann/Keimel/Lawson/ > Mislove/Scott. A little more detail: The opens in the lattice of congruences are determined by the "compacts" of this algebraic lattice. These are the finitely generated congruences. If F is one such, then the open it determines is {E | F subset E}. They form a basis for the "Scott" topology. A subbasis is given by the sets {E | aEb} indicated by Barr. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]