A metric space $(X,d)$ is monotone
if there is a linear order $$ on $X$ and a constant $c$
such that $d(x,y)\leq cd(x,z)$ for all $x y z$ in $X$, and
$\sigma$-monotone if it is a countable union of monotone subspaces.
A planar set homeomorphic to the Cantor set that is not $\sigma$-monotone
is constructed and investigated.
It follows that there is a metric
on a Cantor set that is not $\sigma$-monotone. This answers a question
raised by the second author.
Keywords:
metric space monotone there linear order constant leq sigma monotone countable union monotone subspaces planar set homeomorphic cantor set sigma monotone constructed investigated follows there metric cantor set sigma monotone answers question raised second author
Affiliations des auteurs :
Aleš Nekvinda
1
;
Ondřej Zindulka
1
1
Department of Mathematics Faculty of Civil Engineering Czech Technical University Thákurova 7 160 00 Praha 6, Czech Republic
@article{10_4064_fm213_3_3,
author = {Ale\v{s} Nekvinda and Ond\v{r}ej Zindulka},
title = {A {Cantor} set in the plane that is not $\sigma$-monotone},
journal = {Fundamenta Mathematicae},
pages = {221--232},
year = {2011},
volume = {213},
number = {3},
doi = {10.4064/fm213-3-3},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm213-3-3/}
}
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AU - Ondřej Zindulka
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Aleš Nekvinda; Ondřej Zindulka. A Cantor set in the plane that is not $\sigma$-monotone. Fundamenta Mathematicae, Tome 213 (2011) no. 3, pp. 221-232. doi: 10.4064/fm213-3-3
