Some Properties of Realcompact Subspaces and Coarser Normal Spaces

Abstract

In this work we obtain results in two areas of topology, normal condensations of products and size of realcompact subspaces of a space. In 2000 Swardson proved that every uncountable compact space has a realcompact subspace of the same cardinality as the first uncountable cardinal. In the first chapter the work of Swardson is continued to prove that realcompact spaces with pseudocharacter no greater than the first uncountable cardinal have realcompact subspaces of the same size as the first uncountable cardinal. Under continuum hypothesis, a consequence is that every uncountable realcompact space has a realcompact subspace of the same size as the first uncountable cardinal. We also prove that every realcompact right-separated set of size larger than continuum has a realcompact subspace of size of any cardinal less or equal to continuum. A corollary is that every compact set of size bigger or equal to continuum has a realcompact subset of size less or equal to continuum, answering a question by Professor William Fleissner. In 1997 Buzjakova proved that for a pseudocompact space X, there exists an ordinal such that the product of X and that ordinal condenses onto a normal space if and only if X condenses onto a compact space. In the third chapter, we extend Buzjakovas's method to prove that for a Tychonoff space X, there exists an ordinal such that if the product of X and that ordinal condenses onto a normal space, then X condenses onto a countably paracompact space.