Topics in Set-theoretic Topology

主要调查员将探讨开放的问题，在几个领域ofset-theoretical拓扑。 收敛性在许多领域都有重要的意义，但还没有得到很好的理解。 例如，它是不知道，如果它是一致的空间是Frechet-Urysohn有限集必须是第一可数。 这个问题的解决可能会导致Malychin的著名问题的解决，即可分的Frechet-Urysohn拓扑群必须是度量化的是否是一致的。 这个提议的另外两个主题涉及具有紧开拓扑的函数空间。 解决其中一个问题将解决一个40年的老问题的J. Ceder，解决另一个将产生一个有用的方法来确定这些函数空间是Baire空间。这个建议的另一个目标是应用Balogh的宏伟技术，用于构建ZFC的例子，他用来解决几个长期存在的问题，其他问题，在这个过程中，我们试图简化和编纂这项技术，使其更容易使用。 最后，最近在D-空间性质方面取得了一些进展，这一性质与覆盖性质之间有着有趣但尚未得到很好理解的关系，这表明解决困扰研究人员20多年的货车Douwen的D-空间问题的时机可能已经成熟。集合论和一般拓扑学是数学的基本学科，有着共同的历史根源，它们在数学的许多领域都是必不可少的工具。 一般拓扑学为形状的研究提供了一个框架，从真实的三维空间中的普通形状到更抽象的形状和结构。 例如，连续实值函数是数学的主要内容，因此连续函数空间的拓扑结构在许多领域都有重要的意义，这也是本论文的主题之一。 这同样可以说是关于顺序收敛的概念及其变体，这是本提案的另一个主题。 主要研究者提出的问题是在什么范围内一直是一个富有成效的相互作用一般拓扑学和集合论，一个相互作用所激发的戏剧性进展，在集合论和逻辑在过去四十年左右，并实现许多长期存在的问题，在一般拓扑学的抽象空间休息复杂的集合理论组合。 这些问题的解决方案将加深我们对待探索的基本拓扑性质的理解，并且可能需要新的适用于各种其他问题的集合理论和拓扑技术。

The principle investigator will explore open problems in several areas ofset-theoretic topology. Convergence properties have importantimplications in a number of fields, yet are not well-understood. Forexample, it is not known if it is consistent that a space which isFrechet-Urysohn for finite sets must be first-countable. Settling thisquestion is likely to lead to a solution of Malychin's well-known problemwhether it is consistent that separable Frechet-Urysohn topological groupsmust be metrizable. Two other themes of this proposal involve functionspaces with the compact-open topology. Settling one of them would solve a40 year old problem of J. Ceder, and settling the other would yield auseful way of determining when these function spaces are Baire spaces.Another goal of this proposal is to apply Balogh's magnificent techniquefor constructing ZFC examples , which he used to settle severallong-standing conjectures, to other problems, and in the process attemptto simplify and codify the technique to make it easier to use. Finally,there have been a number of recent advances on the D-space property, aproperty with interesting but not well-understood relationships withcovering properties, which indicate that the time may be ripe for settlingthe D-space questions of van Douwen that have stumped researchers for morethan 20 years. Set theory and general topology are fundamental mathematical disciplines,with common historical roots, and they serve as essential tools in manyareas of mathematics. General topology provides a framework for the studyof shapes, from ordinary shapes in real three-dimensional space to muchmore abstract shapes and structures. For example, continuous real-valuedfunctions are a staple of mathematics, and hence the topological structureof spaces of continuous functions, which is one of the topics that thisproposal centers on, has important implications in many fields. Thesame can be said about the notion of sequential convergence and itsvariants, another topic of this proposal. The principle investigator'sproposed problems lie within the scope of what has been a fruitfulinteraction between general topology and set theory, an interactionspurred by dramatic advances in set theory and logic in the last fortyyears or so and the realization that many long-standing questions in thegeneral topology of abstract spaces rest on complicated set-theoreticcombinatorics. Solutions to the proposed problems would deepen ourunderstanding of the fundamental topological properties to be explored,and would likely require new set-theoretical and topological techniquesapplicable to a variety of other problems.