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