Mathematical Sciences: Problems in Set-Theoretic Topology

集合论拓扑学是拓扑学的一个分支，它用集合论的方法来解决抽象空间的问题。它使用了特殊的公理(如哥德尔的构造性公理、连续统假设、马丁公理和真强迫公理)、强迫技术、内模理论和其他集合论工具来阐明并经常解决长期悬而未决的问题，以及分离和识别值得研究的拓扑性质。调查员将继续他在这一领域的研究。这项研究将侧重于两个主要主题和四个次要主题，根据调查人员和其他人最近的结果，这些主题显示出特别的前景。主要的两个主题是遗传正规性和序列紧性，人们希望最近关于可数紧性主题的成功能够得到推广和平行。第二个主题是：小尺寸的局部紧空间，Frechet-Urysohn拓扑群和向量空间，不可度量流形，以及可数亚紧空间中开集的可数交。

Set-theoretic topology is a branch of topology in which set- theoretic techniques are used to solve problems about abstract spaces. In it, special axioms (such as Godel's Axiom of Constructibility, the Continuum Hypothesis, Martin's Axiom, and the Proper Forcing Axiom), forcing techniques, inner model theory, and other set theoretic tools are used to clarify, and often solve, long-outstanding problems, and to isolate and identify topological properties worthy of study. The investigator will continue his research in this area. The research will focus on two primary topics and four secondary topics which show particular promise in the light of recent results by the investigator and others. The two primary topics are hereditary normality and sequential compactness, where it is hoped that recent successes with the topic of countable tightness can be extended and paralleled. The secondary topics are: locally compact spaces of small size, Frechet-Urysohn topological groups and vector spaces, nonmetrizable manifolds, and countable intersections of open sets in countably metacompact spaces.