Some examples related to colorings

Michael van Hartskamp; Jan van Mill

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 4, page 821-827
  • ISSN: 0010-2628

Abstract

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We complement the literature by proving that for a fixed-point free map f : X → X the statements (1) f admits a finite functionally closed cover 𝒜 with f [ A ] ∩ A = ∅ for all A ∈ 𝒜 (i.e., a coloring) and (2) β 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 ≥ 2 we construct a strongly zero-dimensional Tychonov space X and a fixed-point free map f : X → X such that f admits a closed coloring, but no coloring has cardinality less than n .

How to cite

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van Hartskamp, Michael, and van Mill, Jan. "Some examples related to colorings." Commentationes Mathematicae Universitatis Carolinae 41.4 (2000): 821-827. <http://eudml.org/doc/248609>.

@article{vanHartskamp2000,
abstract = {We complement the literature by proving that for a fixed-point free map $f: X \rightarrow X$ the statements (1) $f$ admits a finite functionally closed cover $\mathcal \{A\}$ with $f[A] \cap A =\emptyset $ for all $A \in \mathcal \{A\}$ (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 $n \ge 2$ we construct a strongly zero-dimensional Tychonov space $X$ and a fixed-point free map $f: X \rightarrow X$ such that $f$ admits a closed coloring, but no coloring has cardinality less than $n$.},
author = {van Hartskamp, Michael, van Mill, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Čech-Stone extension; coloring; Tychonov plank; Čech-Stone extension; coloring; Tikhonov plank},
language = {eng},
number = {4},
pages = {821-827},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some examples related to colorings},
url = {http://eudml.org/doc/248609},
volume = {41},
year = {2000},
}

TY - JOUR
AU - van Hartskamp, Michael
AU - van Mill, Jan
TI - Some examples related to colorings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 4
SP - 821
EP - 827
AB - We complement the literature by proving that for a fixed-point free map $f: X \rightarrow X$ the statements (1) $f$ admits a finite functionally closed cover $\mathcal {A}$ with $f[A] \cap A =\emptyset $ for all $A \in \mathcal {A}$ (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 $n \ge 2$ we construct a strongly zero-dimensional Tychonov space $X$ and a fixed-point free map $f: X \rightarrow X$ such that $f$ admits a closed coloring, but no coloring has cardinality less than $n$.
LA - eng
KW - Čech-Stone extension; coloring; Tychonov plank; Čech-Stone extension; coloring; Tikhonov plank
UR - http://eudml.org/doc/248609
ER -

References

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  1. Aarts J.M., Fokkink R.J., Vermeer J., Variations on a theorem of Lusternik and Schnirelmann, Topology 35 (1996), 4 1051-1056. (1996) MR1404923
  2. Engelking R., General Topology, revised and completed ed., Sigma series in pure mathematics, vol. 6, Heldermann Verlag, 1989. Zbl0684.54001MR1039321
  3. Krawczyk A., Steprāns J., Continuous colourings of closed graphs, Topology Appl. 51 (1993), 13-26. (1993) MR1229497
  4. van Douwen E.K., β X and fixed-point free maps, Topology Appl. 51 (1993), 191-195. (1993) Zbl0792.54037MR1229715
  5. van Hartskamp M.A., Vermeer J., On colorings of maps, Topology Appl. 73 (1996), 2 181-190. (1996) Zbl0867.55003MR1416759
  6. van Mill J., Easier proofs of coloring theorems, Topology Appl. 97 (1999), 155-163. (1999) Zbl0947.54019MR1676678

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