Compact convex sets and cartesian products
by
Ondřej Kalenda
Charles University in Prague

By a compact convex set we mean a compact convex subset of a Hausdorff locally convex space. It is natural to ask which compact convex sets are homeomorphic to a nontrivial product. It follows from Keller's theorem that it is the case for all metrizable compact convex sets except for the singleton and the unit interval. We study this question in the non-metrizable case. There are non-metrizable compact convex sets which are not homeomorphic to a product of two non-metrizable compacta and there are some which are not homeomorphic to a nontrivial product with one factor metrizable. The original question remains open. This question is also related to the problem of homeomorphic classification of non-metrizable compact convex sets.

References:

[1] A.Avilés, O.Kalenda: Fiber orders and compact spaces of uncountable weight, preprint 2007

[2] O.Kalenda: On products with the unit interval, Topol. Appl. 155 (2008), no. 10, 1098-1104; doi:10.1016/j.topol.2008.01.010

Date received: June 23, 2008