From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9529 Path: news.gmane.org!.POSTED!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos theory for spaces of connected components Date: Sun, 4 Feb 2018 12:57:51 -0800 Message-ID: References: Reply-To: John Baez NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1517843189 11254 195.159.176.226 (5 Feb 2018 15:06:29 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Mon, 5 Feb 2018 15:06:29 +0000 (UTC) Cc: categories Original-X-From: majordomo@mlist.mta.ca Mon Feb 05 16:06:25 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 1eiiLR-0001mb-Dw for gsmc-categories@m.gmane.org; Mon, 05 Feb 2018 16:06:09 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:37640) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eiiNF-0002Pg-6K; Mon, 05 Feb 2018 11:08:01 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eiiML-0008Dk-9V for categories-list@mlist.mta.ca; Mon, 05 Feb 2018 11:07:05 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9529 Archived-At: Steve Vickers wrote: > We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components. Digressing a bit, this reminds me of some results David Roberts recently pointed out. However, they concern path-connected components rather than connected components. The set of path-connected components of a space X is a quotient set of X, so we can give it the quotient topology. What can the resulting space be like? Anything! For every topological space X, there is a paracompact Hausdorff space whose space of path-connected components is homeomorphic to X. D. Harris, Every space is a path-component space, Pacific J. Math. 91 no. 1 (1980) 95-104. http://dx.doi.org/10.2140/pjm.1980.91.95 There is more here: https://mathoverflow.net/questions/291443/paths-in-path-component-spaces Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]