From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9530 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos theory for spaces of connected components Date: Mon, 05 Feb 2018 16:12:17 +0000 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit X-Trace: blaine.gmane.org 1517853971 11420 195.159.176.226 (5 Feb 2018 18:06:11 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Mon, 5 Feb 2018 18:06:11 +0000 (UTC) Cc: categories To: John Baez Original-X-From: majordomo@mlist.mta.ca Mon Feb 05 19:06:07 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 1eil9D-00017I-Ll for gsmc-categories@m.gmane.org; Mon, 05 Feb 2018 19:05:43 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:37900) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eilAY-000442-Cw; Mon, 05 Feb 2018 14:07:06 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eil9e-0000dJ-GL for categories-list@mlist.mta.ca; Mon, 05 Feb 2018 14:06:10 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9530 Archived-At: Dear John, For point-set results like this it can be a bit delicate working out how the point-free topos treatment goes. Moerdijk has proved that for a connected, locally connected topos X, the map ends: X^I -> XxX is an open surjection. (Here I = [0,1] is the closed real interval, and if p: I -> X then ends(p) = (p(0), p(1)).) This is interpreted as the appropriate point-free way to say that X is path-connected, so connected, locally connected => path connected - which goes against the classical account. Part of the issue is that a point-free surjection is not necessarily surjective on points. Hence even for locally connected spaces, which are supposed to be the well behaved ones, the path-connected components got from the topos theory (which, by Moerdijk's result, agree with the connected components) may be different from the ones got from point-set topology. All the best, Steve. On 04/02/2018 20:57, John Baez wrote: > 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/ ]