Set-theoretic methods in the study of compact spaces

PI研究了紧空间类中可数收敛问题的独立性结果。基本问题涉及理解无限序列具有收敛连续性或空间包含整数的Stone-Cech紧化的副本的假设之间的二分法。其中一个主要的课题是回答Efimov提出的在每个紧致空间中一个或另一个必须为真是否是一致的问题。另一个是完成我们的项目，确定是否有一个小的上界的自然概念称为序列序的紧凑（序列）空间。整数的Stone-Cech紧化的其余部分（称为N*）的结构，并继Shelah关于自同胚的突破性结果之后继续进行一项重要的研究，即以自同胚为域的连续映射的行为，将被研究。鉴于目前的知识状态，这是相当有趣的，大多数这些问题仍然没有解决的模型，著名的适当的强制公理（PFA）和强制扩展的模型中，这一公理举行。该项目将增加这一中心研究领域的技术和成果工具箱。 毫无疑问，关于N* 的结构有大量的文献，我们将继续我们的项目，研究关于同样基本的空间R*（对于真实的数（半）线R）自然产生的类似问题。有一个根本的区别，产生于这样一个事实，即这个空间是一个连续体和可能的应用拓扑动力学提供额外的动机。

The PI investigates independence results in the study of countableconvergence questions in the class of compact spaces. The fundamental questions concern understanding the dichotomy between the assumption that infinite sequences have converging subsequences or that the space contains a copy of the Stone-Cech compactification of the integers. One of main projects is to answer Efimov's question of whether it is consistent that one or the other must be true in each compact space. Another is to complete our project of determining if there is a small upper bound to the natural notion called the sequential order of a compact (sequential) space. The structure of the remainder of the Stone-Cech compactification of the integers, known as N*, and continuing an important line of investigation following Shelah's breakthrough result on self-homeomorphisms, the behavior of continuous maps with that as the domain, will be investigated. Given the current state of knowledge, it is quite interesting that most of these questions remain unresolved in models of the celebrated Proper Forcing Axiom (PFA) and forcing extensions of models in which this axiom holds. This project will be adding to the toolbox of techniques and results in this central area of study. Unquestionably, there is a huge literature on the structure of N* and we will continue our project of investigating the analogous questions that naturally arise about an equally fundamental space, R* for the real number (half-) line R. There is a fundamental difference that arises from the fact that this space is a continuum and possible applications to topological dynamics provide additional motivation.