On partitions in cylinders over continua and a question of Krasinkiewicz

Mirosława Reńska¹

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Colloquium Mathematicum, Tome 122 (2011) no. 2, pp. 203-214

Résumé

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.

Détail

DOI : 10.4064/cm122-2-5

Keywords: metrizable continuum locally connected only every partition cylinder between bottom top cylinder contains connected partition between these sets krasinkiewicz asked whether every metrizable continuum there exists partiton between top bottom cylinder times hereditarily indecomposable continuum answer question negative present construction partitions continuum which every epsilon admits confluent epsilon mapping locally connected continuum