| 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 |
| . |
