Generalization of Browder's F.P. Theorem?

[Peter McBurney <p.j.mcburney@csc.liv.ac.uk>]

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