Abstract:We complement the literature by proving that for a fixed-point free map $f: X \to X$ the statements (1) $f$ admits a finite functionally closed cover $\Cal A$ with $f[A] \cap A =\emptyset $ for all $A \in \Cal A$ (i.e., a coloring) and (2) $\beta f$ is fixed-point free are equivalent. \par When functionally closed is weakened to closed, we show that normality is sufficient to prove equivalence, and give an example to show it cannot be omitted. \par We also show that a theorem due to van Mill is sharp: for every $n \geq 2$ we construct a strongly zero-dimensional Tychonov space $X$ and a fixed-point free map $f: X \to X$ such that $f$ admits a closed coloring, but no coloring has cardinality less than $n$.
Keywords: \v Cech-Stone extension, coloring, Tychonov plank
AMS Subject Classification: Primary 54G20; Secondary 54C20, 54D15