From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9543 Path: news.gmane.org!.POSTED!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Re: Reflection to 0-dimensional locales Date: Thu, 8 Feb 2018 23:29:39 +0100 Message-ID: References: <5D815D7C26A24888833B8478A002DE64@ACERi3> Reply-To: Andrej Bauer NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1518186264 16608 195.159.176.226 (9 Feb 2018 14:24:24 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 9 Feb 2018 14:24:24 +0000 (UTC) Cc: categories list To: George Janelidze Original-X-From: majordomo@mlist.mta.ca Fri Feb 09 15:24:20 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1ek9ax-0003Cq-Rh for gsmc-categories@m.gmane.org; Fri, 09 Feb 2018 15:24:07 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40187) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ek9cv-0004RH-2d; Fri, 09 Feb 2018 10:26:09 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ek9by-0003qg-4N for categories-list@mlist.mta.ca; Fri, 09 Feb 2018 10:25:10 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9543 Archived-At: On Tue, Feb 6, 2018 at 12:01 PM, George Janelidze wrote: > Dear Colleagues, > > Let me repeat from my exchange of massages with Steve Vickers: > >> As you know, a locale is called 0-dimensional if all its elements are >> joins >> of complemented ones. By a morphism L--->L' of locales I shall mean a map >> L'--->L that preserves all joins and finite meets (as usually). The >> inclusion functor >> >> 0-Dimensional locales--->Locales >> >> has a left adjoint F, for which >> >> F(L)={x in L | x is a join of complementary elements}. >> >> Question: Is F semi-left-exact? Can Example 1 in https://dml.cz/bitstream/handle/10338.dmlcz/119250/CommentatMathUnivCarolRetro_42-2001-2_13.pdf be put to some use to answer the question negatively? It shows that the zero-dimensional reflection in topological spaces does not preserve finite products. The example uses fairly nice subspaces of R and R^2. With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]