Re: Generalization of Browder's F.P. Theorem?

[Michael Barr <barr@barrs.org>]

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
>
>
>
>