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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:
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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