Topics in Set-theoretic Topology

主要研究者将探讨集合论拓扑学的几个领域的开放问题。收敛性质在许多领域都有重要的应用，但还没有得到很好的理解。例如，我们不知道一个有限集为frechet - urysohn的空间是否一定是第一可数的。解决这个问题可能会导致Malychin的著名问题的解决，即可分离的Frechet-Urysohn拓扑群必须是可度量的是否一致。该建议的另外两个主题涉及具有紧开拓扑的功能空间。解决其中一个问题将解决J. Ceder 40年的老问题，解决另一个问题将产生一种确定这些函数空间何时为贝尔空间的有用方法。本提案的另一个目标是将Balogh用于构建ZFC示例的宏伟技术应用于其他问题，他曾使用该技术解决几个长期存在的猜想，并在此过程中尝试简化和编纂该技术以使其更易于使用。最后，最近在d空间性质方面取得了一些进展，d空间性质与覆盖性质之间有着有趣但尚未被很好理解的关系，这表明解决van Douwen的d空间问题的时机可能已经成熟，这些问题已经困扰了研究人员20多年。集合论和一般拓扑是基本的数学学科，具有共同的历史根源，它们在许多数学领域都是必不可少的工具。一般拓扑学为研究形状提供了一个框架，从实际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一个重要组成部分，因此连续函数空间的拓扑结构是本文研究的主题之一，在许多领域都具有重要意义。序贯收敛及其变体的概念也是如此，这是本提案的另一个主题。主要研究者提出的问题是在一般拓扑和集合论之间富有成果的相互作用的范围内，这种相互作用是由过去40年左右集合论和逻辑学的巨大进步以及对抽象空间一般拓扑中许多长期存在的问题依赖于复杂集合论组合的认识所激发的。所提出的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，并且可能需要适用于各种其他问题的新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.