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

集合论拓扑学自上世纪中叶建立以来，一直处于纯数学的基础领域：一般拓扑学和集合论之间。事实上，集合的语言和理论是所有数学公理化的背景，是围绕逻辑、真理和一致性的基本问题被形式化的语言。哥德尔的不完全性定理告诉我们，数学的任何公理基础都将包括独立于公理的陈述(即既不能证明也不能反驳)。事实上，重要的和自然的开放问题被证明是独立的第一个数学领域是拓扑学，这是一个具有几何色彩的数学领域，是分析等数学领域的基本基础。这种独立结果的现象被证明是相当普遍的，甚至是最近在物理学中出现的(证明了一般光谱间隙问题的独立性)。 集合论拓扑学仍然是一个两个领域之间的相互作用促进了技术的进步和发展的领域，这些技术对数学的许多其他领域都有影响和应用。例如，许多重要的组合工具(如强迫技术、Ramsey理论、从Godel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中出现的问题而开发的，后来在泛函分析、群论和组合学等领域得到了广泛的应用。 2007年，M.Hrusak和J.Moore编撰了一份清单《集合论拓扑中的20个问题》，这是该领域最重要和最长期存在的公开问题。这份清单构成了专著《拓扑学中的开放问题》的导言，它遵循了希尔伯特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禧年问题的精神，这些问题旨在鼓励数学界专注于最重要和最有影响力的开放问题并进行合作。我提议的研究计划是围绕赫鲁萨克-摩尔清单中强调的三个相关问题来组织的。Mary Ellen Rudin的问题，是否存在“小的”Dowker空间和van Douwen的问题：Lindelof正则空间是D-空间吗？连续体是紧致齐性空间胞性的界吗？这些问题几十年来一直悬而未决，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列表上的突出地位。这三个问题中的任何一个问题的解决都将涉及发展新的技术和思想，这些技术和思想肯定会在该领域得到应用，并不可避免地应用于数学的其他领域。

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.