Mathematical Sciences: Problems in Set-Theoretic Topology

在某种基本层次上，拓扑学与集合论相结合， 这使它成为数学基础的一部分。 首席研究员将继续研究 集合论拓扑中的几个开放问题。 一个主题是 正规性和可数仿紧性与 局部紧空间 这方面的几个长期问题 保持不稳定，例如，Arhangel'skii-Tall问题（正常 局部紧亚紧空间仿紧？）和沃森的 问题（是否存在一个完全正规的局部紧非- 仿紧空间（ZFC？） 另一个问题导致了有趣的 包含梯形系统的组合学， 更多应用。 其他有待调查的问题包括 道克过滤器的存在，以及沃森和库克的问题 包括连通性。 可能的解决办法包括： 理论公理和组合学，以及强迫。 解决 问题将加深我们对基本拓扑的理解 物业和建筑，并可能需要新的 适用于各种其他问题的技术。

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.