## On partitions in cylinders over continua and a question of Krasinkiewicz

### Volume 122 / 2011

Colloquium Mathematicum 122 (2011), 203-214
MSC: Primary 54F15.
DOI: 10.4064/cm122-2-5

#### Abstract

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 $X\times I$ 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 $\epsilon > 0$, admits a confluent $\epsilon $-mapping onto a locally connected continuum.