From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2106 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Generalization of Browder's F.P. Theorem? Date: Thu, 16 Jan 2003 18:05:39 -0500 (EST) Message-ID: References: <3E26BBF3.2D413C6A@cs.bham.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018415 2514 80.91.229.2 (29 Apr 2009 15:20:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:15 +0000 (UTC) To: CATEGORIES LIST , Original-X-From: rrosebru@mta.ca Fri Jan 17 11:43:40 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Jan 2003 11:43:40 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18ZYcr-0005np-00 for categories-list@mta.ca; Fri, 17 Jan 2003 11:41:45 -0400 X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs X-Sender: barr@triples.math.mcgill.ca In-Reply-To: <3E26BBF3.2D413C6A@cs.bham.ac.uk> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 31 Original-Lines: 76 Xref: news.gmane.org gmane.science.mathematics.categories:2106 Archived-At: In Errett Bishop's Constructive Analysis (anyone who is interested in analysis over a topos absolutely must know this book), he proves that for any continuous endomorphism f of a disk and for every eps > 0, there is a point x in the disk for which |f(x) - x| < eps. A couple of points should be made. First f has to be constructible and eps has to be provably positive. For Bishop, a real number is an equivalence class of pairs of RE sequences of integers a_n and b_n such that for all m,n |a_n/b_n - a_m/b_m| < 1/m + 1/n and a function is constructive if it is a machine for turning one such sequence into another. To be continuous, it there must be a function delta(eps) that produces for each eps > 0, a delta(eps) such that |x - y| < delta(eps) implies that |f(x) - f(y)| < eps. (In fact, there is a non-constructive proof that every constructive function is continuous.) Bishop then claims, without proof as far as I can see, that there is a fixed point free endomorphism of the disk. What this means is that when you extend this function to all reals, any fixed point is not a constructible real number. On Thu, 16 Jan 2003, Steven J Vickers wrote: > > 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 > > > >