Browder’s theorem with general parameter space

It follows from Browder (Summa Bras Math 4:183–191, 1960) that for every continuous function F: (X× Y) → Y, where X is the unit interval and Y is a nonempty, convex, and compact subset of a locally convex linear vector space, the set of fixed points of F, defined by C_F: = { (x, y) ∈ X× Y: F(x, y) = y} , has a connected component whose projection to the first coordinate is X. We extend Browder’s result to the case that X is a connected and compact Hausdorff space.