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