Geodesic
A Cantor set in the plane that is not $\sigma$-monotone
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
Fundamenta Mathematicae, Tome 213 (2011) no. 3, pp. 221-232 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/fm213-3-3
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

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 
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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

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