A Cantor set in the plane that is not $\sigma$-monotone
Tom 213 / 2011
Fundamenta Mathematicae 213 (2011), 221-232 MSC: 54F05, 28A78, 28A80. DOI: 10.4064/fm213-3-3
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.