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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675652
• 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，固定理论拓扑领域已经站在纯数学的基础领域之间的界限：一般拓扑和集合理论。实际上，集合的语言和理论是所有数学被公理化的设置，以及关于逻辑，真理和一致性的基本问题的语言。 Godel的不完整定理告诉我们，任何公理化的数学基础都将包括独立于公理的陈述（即既不可证明也不是可被拒绝的）。实际上，事实证明，重要和自然的开放问题的第一个数学领域是拓扑结构的数学领域，具有几何风味，这是数学领域（例如分析）的基本基础。这种独立结果的现象已被证明是相当内在的，即使是最近在物理学中出现的（证明了一般光谱差距问题的独立性）。 ***固定理论拓扑仍然是一个领域，这两个领域之间的相互作用引起了进步和对许多其他数学领域产生影响和应用的技术的发展。例如，许多重要的组合工具（例如，强迫技术，拉姆西理论，从Godel的可构造宇宙等提取的组合原理等）最初是为了解决拓扑引起的问题而开发的，然后在拓扑引起的问题，然后在潜水员作为功能分析，群体理论和组合学的领域中发现了应用。 ***** 2007年，M。Hrusak和J. Moore汇编了“二十个设定理论拓扑中的问题”，这是该领域最重要，最长期的开放问题。这份清单构成了专着的简介，拓扑的拓扑问题是希尔伯特（Hilbert）在1900年在ICM上形成的著名的100个问题的精神，以及最近的Clay Institute Institute千禧一代问题，旨在鼓励数学社区集中精力并协作最重要和有影响力的开放问题。我提出的研究计划是在Hrusak-Moore列表中突出显示的三个相互联系的问题的组织。玛丽·埃伦·鲁丁（Mary Ellen Rudin）的问题是否存在``小''dowker空间和van douwen的问题：常规空间是dpaces吗？连续体是否结合紧凑型均匀空间的细胞性？这些问题已经开放了数十年，并强调了组合主义者是关于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之间的许多令人惊讶的联系和设定理论问题解释了它们在该领域的重要性以及它们在Hrusak-Moore名单上的重要性。解决这三个问题中的任何一个的解决方案都将涉及开发新技术和思想，这些新技术和思想肯定会在该领域找到应用，并且不可避免地可以在其他数学领域中找到。