A FIXED-POINT free map f : X → X is said to be colorable with k colors if there exists a closed cover ℓ of X consisting of k elements such that C∩f(C) = ∅ for every C in ℓ. 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. Copyright © 1996 Elsevier Science Ltd.
Aarts, J. M., Fokkink, R. J., & Vermeer, H. (1996). Variations on a theorem of Lusternik and Schnirelmann. Topology, 35(4), 1051–1056. https://doi.org/10.1016/0040-9383(95)00057-7