# Article

 Title: Some examples related to colorings (English) Author: van Hartskamp, Michael Author: van Mill, Jan Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 41 Issue: 4 Year: 2000 Pages: 821-827 . Category: math . Summary: 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. 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. 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 $n$. (English) Keyword: Čech-Stone extension Keyword: coloring Keyword: Tychonov plank MSC: 54C20 MSC: 54D15 MSC: 54G20 idZBL: Zbl 1050.54023 idMR: MR1800161 . Date available: 2009-01-08T19:07:46Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119214 . Reference: [1] Aarts J.M., Fokkink R.J., Vermeer J.: Variations on a theorem of Lusternik and Schnirelmann.Topology 35 (1996), 4 1051-1056. MR 1404923 Reference: [2] Engelking R.: General Topology.revised and completed ed., Sigma series in pure mathematics, vol. 6, Heldermann Verlag, 1989. Zbl 0684.54001, MR 1039321 Reference: [3] Krawczyk A., Steprāns J.: Continuous colourings of closed graphs.Topology Appl. 51 (1993), 13-26. MR 1229497 Reference: [4] van Douwen E.K.: $\beta X$ and fixed-point free maps.Topology Appl. 51 (1993), 191-195. Zbl 0792.54037, MR 1229715 Reference: [5] van Hartskamp M.A., Vermeer J.: On colorings of maps.Topology Appl. 73 (1996), 2 181-190. Zbl 0867.55003, MR 1416759 Reference: [6] van Mill J.: Easier proofs of coloring theorems.Topology Appl. 97 (1999), 155-163. Zbl 0947.54019, MR 1676678 .

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