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From: Peter McBurney
Newsgroups: gmane.science.mathematics.categories
Subject: Generalization of Browder's F.P. Theorem?
Date: Wed, 15 Jan 2003 14:00:32 +0000
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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