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

ii In this work we obtain results related to H-sets, Katětov spaces, and H-closed exten-sions 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 |> 2ψ̄(X), where ψ̄(X) is the closed pseudocharacter of the space X. The space provides a coun-terexample 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(Xs), where χ(X) is the character of X and Xs 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 Katětov spaces. We recal