Mathematical Sciences: Problems in Set-Theoretic Topology

集合论拓扑学是利用集合论技术解决抽象空间问题的一个分支。在它中，特殊的公理（如哥德尔的构造性公理、连续统假设、马丁公理和适当强迫公理）、强迫技术、内模型理论和其他集合理论工具被用来澄清，并经常解决长期悬而未决的问题，并分离和识别值得研究的拓扑性质。研究者将继续他在这个领域的研究。研究将集中在两个主要主题和四个次要主题，这些主题在研究者和其他人最近的结果中显示出特别的希望。两个主要主题是遗传正态性和顺序紧密性，希望最近在可数紧密性主题上的成功可以得到扩展和平行。次要的主题是：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，可数元紧空间中开集的可数交集。

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.