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

自上世纪中叶成立以来，集合论拓扑学领域一直处于纯数学的两个基础领域：一般拓扑学和集合论之间。集合的语言和理论，确实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。哥德尔不完备定理告诉我们，任何数学的公理基础都将包含独立于公理的陈述（即既不能证明也不能反驳）。事实上，数学中第一个重要的自然开放问题被证明是独立的领域是拓扑学——一个带有几何色彩的数学领域，是分析等数学领域的基础。这种独立结果的现象已经被证明是相当普遍的，甚至是最近在物理学中出现的（随着一般谱隙问题的独立性的证明）。集合论拓扑学仍然是一个领域，在这个领域中，两个领域之间的相互作用导致了技术的进步和发展，这些技术对许多其他数学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学引起的问题而开发的，然后在功能分析，群论和组合学等不同领域找到了应用。2007年，M. Hrusak和J. Moore编制了一份“集合论拓扑中的20个问题”的清单，这些问题是该领域最重要和长期存在的开放性问题。这个列表构成了专著《拓扑中的开放问题》的引言，它本着希尔伯特1900年在ICM上提出的著名的100个问题和最近的克莱研究所千年问题的精神，旨在鼓励数学界关注和合作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Hrusak-Moore列表中突出的三个相互关联的问题组织的。玛丽·艾伦·鲁丁的问题是否存在一个“小”道克空间和范·道文的问题：林德洛夫正则空间是d空间吗？连续体是紧齐次空间的胞性的a界吗？这些问题已经开放了几十年，并强调了这样一个事实，即组合学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表中的突出地位。这三个问题中的任何一个问题的解决都将涉及到新技术和新思想的发展，这些技术和思想肯定会在本领域得到应用，并不可避免地会应用到其他数学领域。

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.