Browder's Theorem with General Parameter Space

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 \(C_F := \{ (x,y) \in X \times Y \colon F(x,y)=y\}\) 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.