Convergence Properties of Hausdorff Closed Spaces

Abstract

The purpose of this work is to study the topological property of Hausdorff closedness as a purely convergence theoretic property. It is the author's opinion that this perspective proves to be a natural one from which to study H-closedness. Chapter 1 provides a brief introduction to and history of the subject matter. Chapter 2 and the first section of Chapter 3 are mainly preliminary. Here the fundamental facts and definitions needed in the study of H-closed spaces, convergence spaces and especially pretopological spaces are given. In Chapter 2 most proofs are omitted for the sake of brevity, however in Section 3.1, many proofs are given in hopes of helping the reader gain an intuitive feel for pretopologies. Original work begins in Section 3.2, where a study of perfect maps between pretopological spaces is given. Chapters 4 and 5 make up the heart of this work. In Chapter 4, we take an in-depth look at the pretopology $\theta$. This convergence, which can be defined for any topological space, frames both H-closedness and the related property of being an H-set as convergence properties. Upon noting this fact, in Section 4.1, we immediately see the benefits of this framing. Of particular interest to those who have studied H-closed spaces are Theorems 4.1.5 and 4.1.8. Later in this chapter, so-called relatively $\theta$-compact filters, which are defined using the convergence $\theta$, are used to obtain a new characterization of countable Kat\v{e}tov spaces in terms of multifunctions. Chapter 5 provides an analogue of H-closedness which can be defined for any pretopological space. The definition of the so-called PHC spaces is due to the author. In Section 5.1, the PHC spaces are defined and their basic properties are investigated. In Section 5.2, we use the earlier work on perfect maps between pretopological spaces to generate new PHC spaces. Lastly, in Section 5.3, we study the PHC extensions of a pretopological space. In this section we have a construction which is analogous to the Kat\v{e}tov extension of a topological space. Theorem 5.3.6 points to an interesting difference between the usual Kat\v{e}tov extension and our pretopological version. We finish this work with an investigation of the cardinal invariants of pretopological spaces. We are particularly interested in obtaining cardinality bounds for compact Hausdorff pretopological spaces in terms of their cardinal invariants. Throughout the paper we seek to highlight results which distinguish pretopologies from topologies and this chapter features several results of this flavor.