From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7727 Path: news.gmane.org!not-for-mail From: Michael Fourman Newsgroups: gmane.science.mathematics.categories Subject: Re: separable locale Date: Fri, 17 May 2013 12:35:41 +0000 (UTC) Message-ID: References: Reply-To: Michael Fourman NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1368817260 17975 80.91.229.3 (17 May 2013 19:01:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 17 May 2013 19:01:00 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri May 17 21:01:01 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UdPtj-00018z-BC for gsmc-categories@m.gmane.org; Fri, 17 May 2013 21:00:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:57019) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UdPrt-0003Nq-Hl; Fri, 17 May 2013 15:59:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UdPrt-00053e-C7 for categories-list@mlist.mta.ca; Fri, 17 May 2013 15:59:05 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7727 Archived-At: > Prof. Peter Johnstone wrote: >> Dear Thomas, >> >> I'm pretty sure that what Michael meant by "separable" was what >> most topologists would call "second countable" -- i.e., countably >> generated as a frame. (There are some topology textbooks in which >> this condition is called "completely separable".) >> >> Peter Johnstone >> --------------------------------- >> On Tue, 30 Jun 2009, Thomas Streicher wrote: >> >>> Recently rereading Fourman's "Continuous Truth" I came across the term >>> "separable locale" but could nowhere find an explanation. Does it mean a >>> cHa A for which there exists a countable subset B such that ever a in >>> A is >>> the supremum of those b in B with b leq a. This would be the point free >>> account of "second countable", i.e. having a countable basis. >>> Of course, second countable T_) spaces are separable, i.e. have a >>> countable >>> dense set. >>> Is this reading the "usual" one? >>> >>> Thomas > Thomas, Peter is correct about my intention. More precisely, 'separable' is defined in Formal Spaces (FS) (Fourman & Grayson, Brouwer Centenary Symposium) 3.12(c) --- although I now find this account unnecessarily obscure. What you say below is correct classically; constructively there is some subtlety. A locale is separable iff it is presented (as in FS 1.1) by a countable language with decidable \leq and countably many *inhabited* basic covers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]