Certain Weakly Generated Noncompact, Pseudo-compact Topologies on Tychonoff Cubes

Given an uncountable cardinal א, the product space I, I = [0, 1], is called a Tychonoff cube. A collection of closed subsets of a subspace Y of a Tychonoff cube I that covers Y determines a weak topology for Y . The collection of compact subsets of I that have a countable dense subset covers I. 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” subspaces of such I. 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.