On connected partitions in cones
by
Miroslawa Renska
Instytut Matematyki, Uniwersytet Warszawski, Poland

We show that in the cylinder X×I over a non-locally connected metrizable continuum there exists a partition L between the top and the bottom of the cylinder such that L does not contain any connected partition between these sets. Therefore a metrizable continuum Y is locally connected if and only if every partition in the cylinder over Y between the bottom and the top of the cylinder contains a connected partition between these sets. In addition, we show that our arguments extend to the case of some non-metrizable non-locally connected continua.

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. The theorem above shows that in the case of non-locally connected continua we can not construct such partition just repeating the construction of Bing. We present a construction of such partitions for some non-locally connected continua of simple form.

Date received: May 14, 2008