Set-theoretic methods in the study of compact spaces

PI研究紧空间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子序列的假设和空间包含整数的stone - cheh紧化的副本之间的二分法。其中一个主要项目是回答Efimov的问题，即在每个紧化空间中其中一个为真是否一致。另一个是完成我们的项目，确定是否有一个小上界的自然概念，称为紧（顺序）空间的顺序顺序。整数的stone - ech紧化的剩余部分的结构，称为N*，以及继Shelah关于自同态的突破性结果之后的一个重要研究方向，即以其为定域的连续映射的行为，将被研究。鉴于目前的知识状况，有趣的是，在著名的适当强迫公理（PFA）的模型和该公理成立的模型的强迫扩展中，这些问题中的大多数仍未解决。这个项目将增加这一中心研究领域的技术和结果工具箱。毫无疑问，关于N*的结构有大量的文献，我们将继续我们的研究项目，自然地在一个同样基本的空间中出现类似的问题，R*代表实数（半）线R。这个空间是一个连续体，这一事实产生了一个根本的区别，并且可能应用于拓扑动力学提供了额外的动力。

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.