Certain weakly generated noncompact pseudo-compact topologies on Tychonoff cubes

Given an uncountable cardinal ℵ, the product space Iℵ, I=[0,1], is called a Tychonoff cube. A collection of closed subsets of a subspace Y of a Tychonoff cube Iℵ that covers Y determines a weak topology for Y. The collection of compact subsets of Iℵ that have a countable dense subset covers Iℵ. It is known from work of the author and I. Ivanšić that the weak topology generated by this collection is pseudo-compact. We are going to show that it is not compact. The author and I. Ivanšić have also considered weak topologies on some other ``naturally occurring\u27\u27 subspaces of such Iℵ. The new information herein along with the previous examples will lead to the existence of vast naturally occurring classes of pseudo-compacta any set of which has a pseudo-compact product. Some of the classes consist of Tychonoff spaces, so the product spaces from subsets of these are also Tychonoff spaces