A Cantor set in the plane that is not $\sigma$-monotone
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.