标题： 关于D-空间和覆盖性质 显示英文标题

标题： On D-spaces and Covering Properties

摘要： 在本论文中，我们介绍了D-空间及其与众所周知的覆盖性质相关的一些最重要的开放问题。 我们随后介绍了一种新的方法来研究一般情况下的D-空间和覆盖性质。 我们首先在集合$X$的所有主超滤器族上定义了一个拓扑，称为主超滤器拓扑。 我们证明了每个开邻域赋值都可以通过主超滤器拓扑唯一地转换为一个特殊的连续映射。 我们在拓扑范畴中研究了与这个特殊连续映射相关的某些结构，然后通过该映射获得了D-空间的刻画。 最后，我们证明了一些与D性质相关的Lindelöf、仿紧和次紧空间的结果。 显示英文摘要

摘要： 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.