From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2104 Path: news.gmane.org!not-for-mail From: Steven J Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Generalization of Browder's F.P. Theorem? Date: Thu, 16 Jan 2003 14:04:35 +0000 Organization: School of Computer Science, The University of Birmingham, U.K. Message-ID: <3E26BBF3.2D413C6A@cs.bham.ac.uk> References: <3E256980.AEFD39A1@csc.liv.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018414 2505 80.91.229.2 (29 Apr 2009 15:20:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:14 +0000 (UTC) Cc: Peter McBurney To: CATEGORIES LIST Original-X-From: rrosebru@mta.ca Thu Jan 16 18:08:41 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Jan 2003 18:08:41 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18ZI41-0002Hw-00 for categories-list@mta.ca; Thu, 16 Jan 2003 18:00:41 -0400 X-Mailer: Mozilla 4.79 [en] (X11; U; Linux 2.4.18-5 i686) X-Accept-Language: en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 29 Original-Lines: 51 Xref: news.gmane.org gmane.science.mathematics.categories:2104 Archived-At: I'm intrigued by Peter McBurney's question [below]. It looks rather like a question of the constructive content of Brouwer's fixed point theorem. Suppose S is (homeomorphic to) an n-cell. Then in the internal logic of the topos of sheaves over [0,1], f is just a continuous endomap of S. If Brouwer's theorem were constructively true then f would have a fixpoint, and that would come out as a continuous section of the projection [0,1]xS -> [0,1]. More precisely, it would be a map g: [0,1] -> S such that f(x, g(x)) = g(x) for all x. If this existed then the set A = {(x, g(x))| x in [0,1]} would be as required. However, the proof of Brouwer that I've seen is not constructive - it goes by contradiction. So maybe the requirements on A are a way of getting constructive content in Brouwer's result. What is known constructively about Brouwer's fixed point theorem? Steve Vickers. Peter McBurney wrote: > Hello -- > > Does anyone know of a generalization of Browder's Fixed Point Theorem > from R^n to arbitrary topological spaces, or to categories of same? > > ***************** > > Theorem (Browder, 1960): Suppose that S is a non-empty, compact, convex > subset of R^n, and let > > f: [0,1] x S --> S > > be a continuous function. Then the set of fixed points > > { (x,s) | s = f(x,s), x \in [0,1] and s \in S } > > contains a connected subset A such that the intersection of A with {0} x > S is non-empty and the intersection of A with {1} x S is non-empty. > > ***************** > > Many thanks, > > -- Peter McBurney > University of Liverpool, UK