Pseudocompactifications and pseudocompact spaces

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Virginia Polytechnic Institute and State University

We begin this paper with a survey of characterizations of pseudocompact spaces and relate pseudocompactness to other forms of compactness such as light compactness, countable compactness, weak compactness, etc. Some theorems on properties of subspaces of pseudocompact spaces are presented. In particular, conditions are given for the intersection of two pseudocompact spaces to be pseudocompact. First countable pseudocompact spaces are investigated and turn out to be maximally pseudocompact and minimally first countable in the class of completely regular spaces.

We define a pseudocompactification of a space X to be a pseudocompact space in which Xis embedded as a dense subspace. In particular, for a completely regular space X, we consider the pseudocompactification αX = (βX - ζX) U X. We investigate this space and in general all pseudocompact subspaces of βX which contain X. There are many pseudocompact spaces between X and βX, but we may characterize αx as follows: 1) αx is the smallest subspace of βX containing X such that every free hyperreal z-ultrafilter on X is fixed in αx. 2) αx is the largest subspace of βX containing X such that every point in αX - X is contained in a zero set which doesn't intersect X.

The space αx also has the nice property that any subset of X which is closed and relatively pseudocompact in X is closed in αx.

The relatively pseudocompact subspaces of a space are important and are investigated in Chapter 4. We further relate relative pseudocompactness to the hyperreal z-ultrafilter on X and obtain the following characterizations of a relatively pseudocompact zero set: 1) A zero set Z is relatively pseudocompact if and only if Z is contained in no hyperreal z-ultrafilter. 2) A zero set Z is relatively pseudocompact if and only if every countable cover of Z by cozero sets of X has a finite subcover.

In the next chapter we consider locally pseudocompact spaces and obtain results analogous to those for locally compact spaces. Then we relate pseudocompactness and the property of being C* - or C-embedded in a space X. Included in this is a study of certain weak normality properties and their relationship to pseudocompact spaces.

We develop two types of one-point pseudocompactifications and investigate the properties of each. It turns out that a space X is never C* -embedded in its one-point pseudocompactification. Also one space has the property that closed pseudocompact subsets are closed in the one-point pseudocompactification while the other may not have this property but will be completely regular.

We present survey material on products of pseudocompact spaces and unify these results. As an outgrowth of this study we investigate certain functions which are related to pseudocompactness.