Mathematical Sciences: Problems in Set-Theoretic Topology

在一定的基本层面上，拓扑学与集合论结合在一起，使其成为数学基础的一部分。主要研究人员将继续研究集合论拓扑学中的几个未决问题。一个主题是局部紧空间中的正规性和可数仿紧性与仿紧性。这个领域的几个长期存在的问题仍然没有得到解决，例如Arangel‘skii-Tall问题(正规的局部紧的亚紧空间是仿紧的吗？)还有沃森的问题(ZFC中是否存在完全正规的局部紧的非仿紧空间？)另一个问题导致了涉及梯形系统的有趣的组合学，这可能会有更多的应用。其他要调查的问题包括道克过滤器的存在，以及沃森和库克涉及连通性的问题。可能的解决方法包括集合论公理和组合学，以及强迫。这些问题的解决将加深我们对基本拓扑性质和结构的理解，并可能需要适用于各种其他问题的新技术。

At a certain basic level topology merges into set theory, which makes it a part of the foundations of mathematics. The principal investigator will continue his research on several open questions in set-theoretic topology. One theme is normality and countable paracompactness versus paracompactness in locally compact spaces. Several longstanding problems in this area remain unsettled, e.g., the Arhangel'skii-Tall problem (Are normal locally compact metacompact spaces paracompact?) and Watson's problem (Is there a perfectly normal locally compact non- paracompact space in ZFC?) Another problem has led to interesting combinatorics involving ladder systems that will likely have further applications. Other problems to be investigated include the existence of Dowker filters, and questions of Watson and Cook involving connectedness. Possible methods of solution include set- theoretic axioms and combinatorics, and forcing. Solutions to the problems would deepen our understanding of fundamental topological properties and constructions, and would likely require new techniques applicable to a variety of other problems.