On D-spaces and Covering Properties

Abstract

In this thesis, we introduce the subject of D-spaces and some of its most important open problems which are related to well known covering properties. We then introduce a new approach for studying D-spaces and covering properties in general. We start by defining a topology on the family of all principal ultrafilters of a set $X$ called the principal ultrafilter topology. We show that each open neighborhood assignment could be transformed uniquely to a special continuous map using the principal ultrafilter topology. We study some structures related to this special continuous map in the category Top, then we obtain a characterization of D-spaces via this map. Finally, we prove some results on Lindel\"of, paracompact, and metacompact spaces that are related to the property D.