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

### Volume 213 / 2011

Fundamenta Mathematicae 213 (2011), 221-232
MSC: 54F05, 28A78, 28A80.
DOI: 10.4064/fm213-3-3

#### Abstract

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.