Set-theoretic methods and the study of compact spaces

这个项目研究与环境空间中无限集合的收敛或极限性质有关的基本和基本问题。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。这项研究的普遍背景是紧拓扑空间的研究。环境空间中的“点”可以代表从湍流动力系统到量子力学环境的各种应用中的任何数学对象。人们发现，许多自然问题对数学本身的基本原理都很敏感。这项研究项目致力于解决这样的拓扑问题，用现代方法来揭示对数学基本公理的可能依赖。研究人员将研究众所周知的可数性对紧(Hausdorff)空间结构的影响。一些空间是由它们的收敛序列的行为决定的，而最基本的空间之一，从今以后的Betan，整数N的Stone-Cech紧致，处于相反的极端，因为它没有任何收敛序列。Betan的行为很大程度上受到集合论额外公理的影响，许多关于函数空间和度量空间等自然问题也是如此。此外，还将研究区域为Betan紧子集的连续实值函数的Banach空间。同样，集合论假设与附加的拓扑性质混合在一起，导致在重要的紧致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影响它们的集合论公理之间的相互作用。我们期望开发新的技术来构建拓扑空间，以及改进强制技术以揭示所涉及的基本组合结构。这些长期存在的具体问题将是中心焦点。(1)如果无限紧空间没有收敛序列，它是否包含Betan的副本？(2)可度量化紧空间是唯一对角线不可达的空间吗？(3)在集合论的某些模型中，关于剩余Betan-N上的连续函数的Banach空间，像Betan-N本身一样，本质上是唯一的吗？(4)如果紧空间的拓扑是由收敛的可数序列决定的，那么一个人必须迭代多少次才能达到所有这些极限，是否存在一个有限或可数的界？

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?