We show that a metrizable continuum X is locally connected if and only if every partition in the cylinder over X between the bottom and the top of the cylinder contains a connected partition between these sets. J. Krasinkiewicz asked whether for every metrizable continuum X there exists a partiton L between the top and the bottom of the cylinder X × I such that L is a hereditarily indecomposable continuum. We answer this question in the negative. We also present a construction of such partitions for any continuum X which, for every ϵ > 0, admits a confluent ϵ -mapping onto a locally connected continuum.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-2-5,
author = {Miros\l awa Re\'nska},
title = {On partitions in cylinders over continua and a question of Krasinkiewicz},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {203-214},
zbl = {1219.54037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-2-5}
}
Mirosława Reńska. On partitions in cylinders over continua and a question of Krasinkiewicz. Colloquium Mathematicae, Tome 122 (2011) pp. 203-214. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-2-5/