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，包括保罗·J·科恩开创的强迫方法，以及对集合论和大基数内部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。同时，紧Hausdorff空间的理论也取得了显著的进展，而且并不总是很容易预先看出是否需要额外的集合论假设。这位研究者打算继续他对紧Hausdorff空间的研究，包括特殊集合论工具的应用。其他受到关注的空间将是相关的可数紧空间和序列紧空间，Moore空间，不可度量流形，Frechet-Urysohn拓扑群和拓扑向量空间，可数亚紧空间，以及Banach空间的弱和弱*紧的子集。正规空间上的有限群作用也将受到一些关注。一般拓扑学比其他拓扑学更是一门基础性学科，它本身涉及非常抽象的空间，以及采用集合论公理的变体对拓扑学的影响。尽管与熟悉的几何对象的类比是有价值的启发式方法，但涉及低维流形的可塑性、直观的几何与Banach空间的子集的研究相去甚远，后者与经典分析的关系比与几何更密切，尽管使用了几何术语。尼科斯十年前参与的一般拓扑学的一个主要结果是发现，关于摩尔空间的一个著名的未经证明的猜想实际上独立于通常的集合论基本公理。此外，它被证明与康托连续统假说不一致，这是一个更熟悉的命题，即如果必须做出选择，大多数在职数学家很可能会选择。这个结果暗示了最佳一般拓扑的味道。

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.