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?
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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.
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Many thanks,
-- Peter McBurney University of Liverpool, UK