Mathematical Sciences: Compact Hausdorff Spaces and Set-Theoretic Topology

在过去的三十年里，由于集合论中强大的新技术，包括Paul J. Cohen开创的强迫方法，以及对集合论和大基数的内模型结构的深入研究，一般拓扑学和布尔代数理论发生了一场革命。与此同时，紧化Hausdorff空间的理论也取得了惊人的进展，但要事先判断是否需要额外的集合论假设并不总是容易的。研究者打算继续他的紧Hausdorff空间的研究，包括特殊集合论工具的应用。其他需要注意的空间类别将是可数紧致和顺序紧致空间，摩尔空间，非度量流形，Frechet-Urysohn拓扑群和拓扑向量空间，可数元紧致空间，以及Banach空间的弱和弱*紧致子集的相关类别。正常空间上的有限群作用也将得到一些关注。一般拓扑学比其他拓扑学更像是一门基础学科，它涉及非常抽象的空间，以及集合论公理变体的采用对拓扑学的影响。尽管与熟悉的几何对象的类比是有价值的启发式方法，但涉及低维流形的塑性直观几何与巴拿赫空间子集的研究相距甚远，后者与经典分析的关系比几何更密切，尽管使用了几何术语。十年前Nyikos参与的一般拓扑学的一个主要成果是发现了一个关于摩尔空间的著名的未被证明的猜想实际上是独立于集合论的通常基本公理的。此外，它被证明与康托连续统假设不一致，康托连续统假设是一个更熟悉的命题，大多数工作的数学家如果必须做出选择，他们可能会选择康托连续统假设。这一结果表明了最佳通用拓扑的特点。

Over the last three decades there has been a revolution in general topology and the theory of Boolean algebras, brought about by powerful new techniques in set theory, including the method of forcing pioneered by Paul J. Cohen, and by deep studies in the structure of inner models of set theory and large cardinals. At the same time, there has been striking progress in the theory of compact Hausdorff spaces, and it is not always easy to see in advance whether extra set-theoretic assumptions are required. The investigator intends to continue his research on compact Hausdorff spaces, including the application of special set-theoretic tools. Other classes of spaces receiving attention will be the related classes of countably compact and sequentially compact spaces, Moore spaces, non-metrizable manifolds, Frechet-Urysohn topological groups and topological vector spaces, countably metacompact spaces, and weak and weak* compact subsets of Banach spaces. Finite group actions on normal spaces will also receive some attention. General topology is more of a foundational subject than the rest of topology, concerning itself with very abstract spaces and with the impact on topology of the adoption of variants of the axioms of set theory. Although analogies to familiar objects of geometry are valuable heuristics, the plastic, intuitive kind of geometry involving low-dimensional manifolds is a far cry from the study of subsets of a Banach space, the latter being more closely related to classical analysis than to geometry, in spite of the geometric terminology employed. One major result of general topology in which Nyikos was involved a decade ago was the discovery that a famous unproved conjecture concerning Moore spaces was actually independent of the usual basic axioms of set theory. Moreover, it was shown to be inconsistent with the Cantor Continuum Hypothesis, a far more familiar proposition that most working mathematicians would be likely to choose if they had to make a choice. This result suggests the flavor of the best general topology.