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。它利用特殊的公理(如哥德尔的构造性公理、马丁公理、覆盖引理和乘积测度扩张公理)、强迫技巧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，并通过一致性结果来分离和识别值得研究的拓扑性质。首席研究员将继续他在这一分支的研究，集中在四个领域，最近的结果极大地提高了我们对所讨论的空间的理解，在某些情况下，还包括集合论工具本身。所讨论的空间是各种类型的可数紧空间；Frechet链网空间；不可度量流形；以及可数亚紧空间。这样的研究可能不会像更多的几何拓扑学那样直接应用，但从长远来看，可以预期影响我们对空间的想法，甚至关于证明的性质。

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, Martin's Axiom, the covering Lemma, and the Product Measure Extension Axiom), forcing techniques, and other set-theoretic tools are used to clarify, and often solve, long- outstanding problems, and through consistency results to isolate and identify topological properties worthy of study. The principal investigator will continue his research in this branch, centered on four areas where recent results have greatly enhanced our understanding of the spaces in question and, in some cases, the set-theoretic tools themselves. The spaces in question are various classes of countably compact spaces; Frechet chain net spaces; nonmetrizable manifolds; and countably metacompact spaces. Such studies will probably have less immediate application than more geometric topological studies but in the long run can be expected to influence our ideas about space and even about the nature of proof.