Some problems from set-theoretic topology - normality, D-spaces and homogeneity

集合论拓扑的一些问题 - 正态性、D 空间和同质性

• 批准号: RGPIN-2019-06356
• 负责人: Szeptycki, Paul
• 金额: $ 1.09万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=714792
• 关键词: Some; problems; set; theoretic; topology

Since its establishment in the middle of the last century, the field of set-theoretic topology has stood on the boundary between to foundational areas of pure mathematics: General Topology and Set Theory. The language and theory of sets is, indeed, the setting in which all of mathematics is axiomatized and the language in which fundamental questions around logic, truth and consistency are formalized. Godel's incompleteness theorem tells us that any axiomatic foundation for mathematics will include statements independent of the axioms (i.e., neither provable nor refutable). And indeed, the first area of mathematics where important and natural open problems turned out to be independent was in Topology an area of mathematics with a geometric flavour that is fundamental basis for areas of mathematics such as Analysis. This phenomenon of independence results has turned out to be quite endemic, even quite recently arising in Physics (with the proof of the independence of the general spectral gap problem). Set-theoretic topology is still an area where the interplay between both areas give rise to advances and the development of techniques that have impact and applications to many other areas of mathematics. For example, many important combinatorial tools (e.g., forcing techniques, Ramsey theory, combinatorial principles extracted from Godel's constructible universe, etc) that were initially developed to solve problems arising from Topology then found applications in areas as diverse as Functional Analysis, Group Theory and combinatorics. In 2007, M. Hrusak and J. Moore compiled a list “Twenty problems in set-theoretic topology” the most important and long-standing open problems in the field. This list, which formed the introduction to the monograph, Open Problems In Topology, was in the spirit of Hilbert's famous 100 problems formulated at the ICM in 1900 and the more recent Clay Institute Millennial problems, which were meant to encourage the mathematical community to focus and collaborate on the most important and impactful open problems. My proposed program of study is organized around three connected problems highlighted in the Hrusak-Moore list. Mary Ellen Rudin's problem whether there is a ``small'' Dowker space and the problems of van Douwen: Are Lindelof regular spaces D-spaces? And is the continuum the a bound on the cellularity of compact homogeneous spaces? These problems have been open for many decades and underscore the fact that combinatorics lie at the heart of many fundamental questions about the structure of topological spaces. Moreover, the many surprising connections between these problems and other topological and set theoretic questions explain their importance in the field and their prominence on the Hrusak-Moore list. The solution of any of these three problems will involve the development of new techniques and ideas which are certain to find applications in the field and inevitably to other areas of mathematics.

自上世纪中叶建立以来，集合论拓扑领域一直处于纯数学基础领域：一般拓扑学和集合论之间的边界。事实上，集合的语言和理论是所有数学被公理化的环境，也是围绕逻辑、真理和一致性的基本问题被形式化的语言。哥德尔不完备性定理告诉我们，任何数学公理基础都将包括独立于公理的陈述（即既不可证明也不可反驳）。事实上，第一个重要且自然的开放问题被证明是独立的数学领域是拓扑学，一个具有几何风格的数学领域，它是分析等数学领域的基础。这种独立结果的现象已被证明是相当普遍的，甚至最近在物理学中也出现了（并证明了一般光谱间隙问题的独立性）。 集合论拓扑仍然是一个领域，两个领域之间的相互作用促进了技术的进步和发展，这些技术对数学的许多其他领域产生了影响和应用。例如，许多重要的组合工具（例如，强迫技术、拉姆齐理论、从哥德尔的可构造宇宙中提取的组合原理等）最初是为了解决拓扑问题而开发的，后来在泛函分析、群论和组合学等不同领域得到了应用。 2007 年，M. Hrusak 和 J. Moore 编制了一份“集合论拓扑中的二十个问题”列表，列出了该领域最重要且长期存在的开放问题。这份清单构成了《拓扑学中的开放问题》专着的引言，体现了 1900 年在 ICM 上提出的希尔伯特著名的 100 个问题和最近的克莱研究所千禧年问题的精神，这些问题旨在鼓励数学界关注和合作解决最重要和最有影响力的开放问题。我提出的研究计划是围绕赫鲁萨克-摩尔清单中突出显示的三个相互关联的问题组织的。 Mary Ellen Rudin 的问题是否存在“小”Dowker 空间以及 van Douwen 的问题：Lindelof 正则空间是 D 空间吗？连续统是致密同质空间的细胞结构上的界限吗？这些问题已经存在了几十年，并强调了这样一个事实：组合数学是有关拓扑空间结构的许多基本问题的核心。 此外，这些问题与其他拓扑和集合论问题之间的许多令人惊讶的联系解释了它们在该领域的重要性以及它们在 Hrusak-Moore 列表中的突出地位。 这三个问题中任何一个的解决都将涉及新技术和新思想的发展，这些新技术和新思想肯定会在该领域得到应用，并且不可避免地会应用于数学的其他领域。