Fixed-point free maps of Euclidean spaces

Our main result states that every fixed-point free continuous self-map of ${\mathbb{R}}^n$ is colorable. This result can be re-formulated as follows: A continuous map $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is fixed-point free iff $\tilde{f}:\beta {\mathbb{R}}^n\to \beta {\mathbb{R}}^n$ is fixed-point free. We also obtain a generalization of this fact and present some examples.