A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-3-3,
author = {Ale\v s Nekvinda and Ond\v rej Zindulka},
title = {A Cantor set in the plane that is not $\sigma$-monotone},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {221-232},
zbl = {1227.54037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-3-3}
}
Aleš Nekvinda; Ondřej Zindulka. A Cantor set in the plane that is not σ-monotone. Fundamenta Mathematicae, Tome 215 (2011) pp. 221-232. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-3-3/