Mathematical Sciences: Compact Hausdorff Spaces and Set-Theoretic Topology

在过去的三十年里， 一般拓扑学和布尔代数理论，带来了 集合论中强大的新技术，包括 由Paul J. Cohen开创，并通过深入研究 集合论内部模型的结构和大基数。 在 与此同时，在理论上也取得了显著的进展， 紧的Hausdorff空间，并且它并不总是容易看到， 提出是否需要额外的集合论假设。 的 调查人员打算继续他对紧凑型豪斯多夫的研究 空间，包括特殊的集合论工具的应用。 其他受到关注的空间类别将与 可数紧和序列紧空间类，摩尔 空间，不可度量化流形，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 * 紧子集。 有限群 在正常空间上的动作也会受到一些关注。 一般拓扑学是一门基础学科， 拓扑学的其余部分，涉及非常抽象的空间， 随着采用的变体的拓扑结构的影响， 集合论的公理 虽然类比熟悉的对象， 几何学是有价值的数学，是可塑的，直观的， 涉及低维流形的几何与 研究子集的Banach空间，后者更密切 与经典分析相比，与几何相比，尽管 使用的几何术语。 一般的一个主要结果 Nyikos十年前参与的拓扑结构是 发现一个关于摩尔空间的著名的未经证明的猜想 实际上是独立于集合论的基本公理的。 此外，它被证明是不符合康托连续统 假设，一个更为熟悉的命题， 如果数学家们必须做出一个 选择 这一结果表明，最好的一般 topology.

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.