Study of Countable Convergence Conditions in Compact Spaces

主要研究者提出集中研究集合论拓扑中的三个核心问题，涉及基本可数收敛问题和集合论独立性结果。第一个问题，由于Efimov，是确定它是否一致，任何紧化空间，不包含收敛序列将包含整数的Stone-Cech紧化。第二，由于Bashkirov，确定紧序空间的序阶上是否存在可数界。在某些方面，这个问题可以被看作是最近解决的摩尔-莫罗卡问题的一个非常有趣的强化。第三种方法是继续对实数的石切赫余数进行系统的研究，类似于长期以来对整数余数所做的研究。由于连通性属性所带来的额外复杂性，似乎存在许多阻碍进展的障碍。收敛序列的存在性和收敛序列在多大程度上完全决定拓扑结构的一般问题在各种情况下经常出现。这些问题在函数空间的紧子集（实值函数集合上的自然信息拓扑）的背景下特别有意义。研究人员正在继续进行研究，这些研究刺激了一般拓扑学和集合论领域之间的密切互动

DMS-0103985Alan DowThe Principle Investigator proposes to focus on three central problems in set-theoretic topology which involve basic countable convergence issues and set-theoretic independence results. The first question, due to Efimov, is to determine if it is consistent that any compact space that does not contain a converging sequence will contain the Stone-Cech compactification of the integers. The second, due to Bashkirov, is to determine if there can be a countable bound on the sequential order of a compact sequential space. This problem can, in some ways, be viewed as a very interesting strengthening of the recently resolved Moore-Mrowka problem. The third is to continue a systematic study of the Stone-Cech remainder of the reals analogous to that which has long been conducted for the remainder of the integers. There appears to be many obstructions to progess caused by the additional complexity imposed by the connectedness property.General questions of the existence of converging sequences and the extent to which converging sequences fully determine the topological structure arise frequently in a variety of settings. These questions are particularly meaningful in the context of compact subsets of function spaces (naturalinformative topologies on sets of real-valued functions). The investigator is continuing to pursue lines of investigation that have stimulated intensive interaction between the fields of general topology and set-theory for de