On products with the unit interval (Q929993)

It is known that there are certain compact convex sets that are not homeomorphic to a product of two non-metrizable compacta. This is true for example for the space \(P(A(\kappa))\) of Radon probability measures on the one-point compactification of an uncountable discrete space endowed with the weak\(^*\) topology. It is natural to ask whether these examples are homeomorphic to a nontrivial product where one of the factors is metrizable. In this interesting paper, the author proves that for an infinite compact scattered space \(K\) we have that \(P(K)\) and \(P(K)\times [0,1]\) are homeomorphic. For other examples of compact convex sets that are not products of two non-metrizable compacta this is not known. For example, this is not known for \(B(\kappa)\), the unit ball of the non-separable Hilbert space \(\ell_2(\kappa)\) with the weak topology. The paper ends with several open problems. For example, the question whether \(\{0\}\) and \([0,1]\) are the only compact convex sets that are not homeomorphic to any nontrivial product.