From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3920 Path: news.gmane.org!not-for-mail From: "Robert J. MacG. Dawson" Newsgroups: gmane.science.mathematics.categories Subject: Re: Stupid question: what space was Euclid working in? (almost) Date: Tue, 18 Sep 2007 09:36:10 -0300 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 X-Trace: ger.gmane.org 1241019605 10894 80.91.229.2 (29 Apr 2009 15:40:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:05 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 18 20:33:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Sep 2007 20:33:45 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXmPF-0005VM-VX for categories-list@mta.ca; Tue, 18 Sep 2007 20:23:02 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 44 Original-Lines: 19 Xref: news.gmane.org gmane.science.mathematics.categories:3920 Archived-At: tporter@informatics.bangor.ac.uk wrote: > If one assumes that `region' is a more basic notion of position than > `point' a lot of Euclid still goes through but dimension seems very hard > to handle. The topological form of Helly's theorem might be a place to start. However, defining dimension in terms of _convex_ structure is very tricky once you get into general spaces. For instance, I showed in my thesis (in a section eventualLy rewritten for _Cahiers_) that if you define a "convex set" on S^1 to be an arc shorter than an open semicircle, you get the obvious homology. However, if closed semicircles and their intersections are convex, the homology becomes that of the 2-sphere. -Robert