Variations on a theorem of lusternik and schnirelmann

A fixed-point free map f: X → X is said to be colorable with k colors if there exists a closed cover β of X consisting of k elements such that C∩f(C) = φ for every C in β. It is shown that each fixed-point free involution of a paracompact Hausdorff space X with dim X ≤ n can be colored with n + 2 colors. Each fixed-point free homeomorphism of a metrizable space X with dim X ≤ n is colorable with n + 3 colors. Every fixed-point free continuous selfmap of a compact metrizable space X with dim X ≤ n can be colored with n + 3 colors