Generalization of Browder's F.P. Theorem?

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