Set-theoretic methods and the study of compact spaces

集合论方法和紧空间的研究

• 批准号: 1501506
• 负责人: Alan Dow
• 金额: $ 18万
• 依托单位: University of North Carolina at Charlotte
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501506&HistoricalAwards=false
• 关键词: Set; theoretic; methods; study; compact

This project investigates fundamental and foundational questions in connection with convergence or limiting properties of infinite sets in an ambient space. The issues of the long-term behavior of an infinite sequence of processes or systems (however abstract) is at the core of many mathematical or physical questions. The universal setting for this investigation is known as the study of compact topological spaces. The "points" in the ambient space can represent any mathematical object in applications ranging from turbulent dynamical systems to a quantum mechanics setting. It has emerged that many natural questions are sensitive to the very foundational principles of mathematics itself. This research project pursues the resolution of such topological questions armed with the modern methods of uncovering possible dependence on the foundational axioms of mathematics. The investigator will study the influence of well-known countability properties on the structure of compact (Hausdorff) spaces. Some spaces are determined by the behavior of their converging sequences, while one of the most fundamental spaces, henceforth BetaN, the Stone-Cech compactification of the integers N, are at the opposite extreme in that it has no converging sequences whatsoever. The behavior of BetaN is heavily influenced by extra axioms of set-theory and many natural questions about spaces of functions and measures and the like, are as well. Additionally, the Banach spaces of continuous real-valued functions with domain some compact subset of BetaN will be investigated. Again set-theoretic hypotheses, mixed with additional topological properties, lead to varying behavior in the important class of compact spaces. Our study is the analysis of the interplay between the various countable convergence conditions and the axioms of set-theory that influence them. We are expecting to have to develop new techniques of constructing topological spaces as well as the refinement of forcing techniques to uncover basic combinatorial structures involved. These specific problems of longstanding will be of central focus. (1) If an infinite compact space has no converging sequence, does it contain a copy of BetaN? (2) Are the metrizable compact spaces the only ones whose diagonal is uncountably-inaccessible? (3) Is the Banach space of continuous functions on the remainder BetaN - N, like BetaN - N itself, essentially unique in certain models of set theory? (4) If the topology of a compact space is determined by converging countable sequences, is there a finite or countable bound on how many times one must iterate taking limits to capture all such limits?

本项目研究与周围空间中无限集合的收敛或极限性质有关的基本和基础问题。无限序列的过程或系统（无论多么抽象）的长期行为问题是许多数学或物理问题的核心。这种研究的普遍背景被称为紧拓扑空间的研究。周围空间中的“点”可以表示从湍流动力系统到量子力学设置的应用中的任何数学对象。许多自然问题对数学本身的基本原理都很敏感。这个研究项目追求这样的拓扑问题的解决武装与现代方法发现可能依赖于数学的基本公理。研究者将研究著名的可数性质对紧致（豪斯多夫）空间结构的影响。有些空间是由它们的收敛性决定的 序列，而最基本的空间之一，此后BetaN，整数N的Stone-Cech紧化，处于相反的极端，因为它没有收敛序列。BetaN的行为受到集合论的额外公理的严重影响，许多关于函数和测度等空间的自然问题也是如此。此外，还研究了定义域为BetaN的某个紧子集的连续实值函数的Banach空间。再一次，集合论的假设，加上额外的拓扑性质，导致了在重要的紧空间类中的不同行为。我们的研究是分析各种可数收敛条件和集合论公理之间的相互作用，影响他们。 我们期望必须开发新的技术来构建拓扑空间，以及改进强制技术来揭示所涉及的基本组合结构。这些长期存在的具体问题将成为中心焦点。(1)如果一个无限紧空间没有收敛序列，它是否包含BetaN的拷贝？(2)可度量化紧空间是唯一对角线不可达的空间吗？(3)是Banach空间的连续函数的其余部分BetaN-N，像BetaN-N本身，本质上是唯一的某些模型的集合论？(4)如果一个紧空间的拓扑是由收敛的可数序列决定的，那么对于必须求极限多少次才能得到所有的极限，是否存在一个有限的或可数的界限？