Browder's Theorem with General Parameter Space

Abstract

Browder (1960) proved that for every continuous function $F:X\times Y\to Y$, where $X$ is the unit interval and $Y$ is a nonempty, convex, and compact subset of $\dR^n$, 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 to the first coordinate is $X$. We extend this result to the case where $X$ is a connected and compact Hausdorff space.