Set-theoretic methods in the study of compact spaces

PI研究紧空间类中可数收敛问题的独立性结果。基本问题涉及到理解假设无限序列具有收敛的子序列或空间包含整数的Stone-Cech紧凑化的副本之间的二分法。主要项目之一是回答埃菲莫夫的问题，即在每个紧凑的空间中，一个或另一个肯定是正确的，这一问题是否一致。另一个是完成我们的项目，即确定自然概念是否存在一个小的上界，称为紧凑(顺序)空间的顺序。我们将研究整数的Stone-Cech紧化的剩余部分(称为N*)的结构，并在Shelah关于自同胚的突破性结果之后继续进行重要的研究，即以此为域的连续映射的行为。鉴于目前的知识状况，相当有趣的是，这些问题中的大多数在著名的真强迫公理(PFA)和强迫扩展的模型中仍然没有得到解决，在这些模型中，这个公理是成立的。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*对于实数(半)线R自然产生的类似问题。这个空间是一个连续体，这一事实产生了根本的不同，可能的拓扑动力学应用提供了额外的动机。

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.