Periodic points, compactifications and eventual colorings of maps

In this paper, we investigate some properties concerning periodic points of maps and compactifications of spaces. By use of the properties, we prove the following theorems:(i) Let X   be a finite-dimensional separable metric space and let f:X→Xf:X→X be a fixed-point free closed map with zero-dimensional set of periodic points. If f:X→Xf:X→X satisfies the conditionsup{|f−1(x)|;x∈X}<∞, then f is eventually 2-colorable.(ii) Let X   be a locally compact, separable metric finite-dimensional space. If f:X→Xf:X→X is any fixed-point free map with zero-dimensional set of periodic points, then f is eventually 2-colorable.