On D-spaces and Covering Properties
Talal Alrawajfeh, Hasan Z. Hdeib
TL;DR
This work addresses long-standing open questions about D-spaces and their relationship to classical covering properties by introducing a novel, mapping-centered framework. Central to the approach is the principal ultrafilter topology, which converts open neighborhood and covering assignments into continuous maps (companions), enabling a new, equivalent definition of D-spaces via kernels and pullbacks. The core contributions include a comprehensive characterization of paracompactness and metacompactness through companion maps, a mapping-driven formulation of D-spaces, and results connecting Lindelöf, paracompact, and metacompact D-spaces within this framework. While the puf-based method provides fresh insights and potential pathways to resolving the Lindelöf and paracompact questions, the author also discusses limitations and avenues for refining the topology and extending the approach to broader covering properties. Overall, the paper offers a unifying, category-theoretic lens for studying D-spaces, highlighting both the promise and challenges of a mapping-centric program in topology.
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 Lindelöf, paracompact, and metacompact spaces that are related to the property D.