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On partitions in cylinders over continua and a question of Krasinkiewicz

Mirosława Reńska — 2011

Colloquium Mathematicae

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...

On Cantor manifolds for the small transfinite dimension

Mirosława Reńska — 2006

Commentationes Mathematicae

We show that some of the spaces X α constructed in [7] are Cantor manifolds for the small transfinite dimension trind, which is transfinite extension of the Menger-Urysohn dimension. That gives us the construction of such spaces that is simpler than constructions of metrizable Cantor manifolds for trind published hitherto. In addition, our examples are disjoint unions of Euclidean cubes and the irrationals.

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