Browder's Theorem: from One-Dimensional Parameter Space to General Parameter Space

Abstract

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f:(X\times Y)\to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff topological space and $Y$ is a nonempty, convex, and compact subset of a locally-convex topological vector space, the set of fixed points of $f$, defined by $C_f:=\{(x,y)\in X\times Y:f(x,y)=y\}$, has a connected component whose projection onto the first coordinate is $X$. In this note we provide an elementary proof for this result, using its reduction to the case $X=[0,1]$.