EN
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.