PROPERTIES OF H-SETS, KATETOV SPACES AND H-CLOSED EXTENSIONS WITH COUNTABLE REMAINDER

Abstract

In this work we obtain results related to H-sets, Katetov spaces and H-closed extensions with countable remainder. As we shall see, these three areas are closely related but the results of each section carry their own definite flavor. Our first results concern finding cardinality bounds of H-sets in Urysohn spaces. In particular, a Urysohn space X is constructed which has an H-set A with |A| > 2 2ψ(X), where ψ(X) is the closed pseudocharacter of the space X. The space provides a counterexample to Fedeli's question in [16]. In addition, it is demonstrated that there is no θ-continuous map from a compact Hausdorff space to the space X with the H-set A as the image, giving a Urysohn counterexample to Vermeer's conjecture in [51]. Finally, it is shown that the cardinality of an H-set in a Urysohn space X is bounded by 2χ(X(s)), where χ(X) is the character of X and X(s) is the semiregularization of X. This refines Bella's result in [4] that the cardinality of such an H-set is bounded by 2χ(X). The next section concerns the relationship of H-sets and Katetov spaces. We recall that a Katetov space can be embedded as an H-set in some space. Herrlich showed in [23] that the space of rational numbers, Q, is not Katetov. Later Porter and Vermeer [41] refined this result with the fact that countable Katetov spaces are scattered. We obtain a similar refinement of Herrlich's result, and a generalization under an additional set-theoretic assumption. Our results include that a countable crowded space cannot be embedded as an H-set and that, assuming the Continuum Hypothesis, neither can the minimal η1 space. Chapter 4 investigates necessary and sufficient conditions for a space to have an H-closed extension with countable remainder. For countable spaces we are able to give two characterizations of those spaces admitting an H-closed extension with countable remainder. The general case appears more difficult, however, we arrive at a necessary condition - a generalization of Cech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular, using the notation of Csaszar in [11], we show that a space X is a Cech g-space if and only if X is Gδ in σX or equivalently if EX is Cech complete. An example of a space which is a Cech f -space but not a Cech g-space is given answering a couple of questions of Csaszar. We show that if X is a Cech g-space and R(EX), the residue of EX, is Lindelof, then X has an H-closed extension with countable remainder. Finally, we investigate some natural extensions of the residue to the class of all Hausdorff spaces.