Browder's Theorem through Brouwer's Fixed Point Theorem

A parametric version of Brouwer's fixed point theorem, called Browder's theorem, states that for every continuous mapping f:[0,1]xX & RARR;X , where X is a nonempty, compact, and convex set in a Euclidean space, the set of fixed points of f, namely, the set {(t,x)& ISIN;[0,1]xX:f(t,x)=x} , has a connected component whose projection onto the first coordinate is [0,1] . Browder's original proof relies on the theory of the fixed point index. We provide an alternative proof that uses Brouwer's fixed point theorem and is valid whenever X is a nonempty, compact, and convex subset of a Hausdorff topological vector space.