From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6434 Path: news.gmane.org!not-for-mail From: Dana Scott Newsgroups: gmane.science.mathematics.categories Subject: Re: Does this topology have a name? Date: Sat, 1 Jan 2011 10:53:55 -0800 Message-ID: References: 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 1293980603 28683 80.91.229.12 (2 Jan 2011 15:03:23 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 2 Jan 2011 15:03:23 +0000 (UTC) Cc: Categories list To: Michael Barr Original-X-From: majordomo@mlist.mta.ca Sun Jan 02 16:03:18 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 1PZPSo-0004cC-IL for gsmc-categories@m.gmane.org; Sun, 02 Jan 2011 16:03:18 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:50729) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PZPSQ-0000XE-Gy; Sun, 02 Jan 2011 11:02:54 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PZPSL-0001DI-8o for categories-list@mlist.mta.ca; Sun, 02 Jan 2011 11:02:49 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6434 Archived-At: On Jan 1, 2011, at 6:59 AM, Michael Barr wrote: > Let A be a model of a finitary equational theory and let X be the set of > congruences on A. For a,b in A, let M(a,b) = {E} such that E is a > congruence on A and aEb. Does this topology have a name? It turns out > that this topology is coherent which means, among other things, that if we > make the M(a,b) clopen, the result is a Stone space. Consider the powerset space P(A x A) = 2^(A x A). The product topology makes it a Stone space. This is elementary. Now, the space X of congruences is defined by logical formulae with only universal quantifiers and atomic formulae xEy for variables ranging over A. That makes X a CLOSED subspace of P(A x A). This is so easy, it hardly needs a name. And it works even if A has infinitely many operations. That there is an equational theory in the background seems neither here nor there to get a Stone space of congruences. HAPPY NEW YEAR! [For admin and other information see: http://www.mta.ca/~cat-dist/ ]