Study of Countable Convergence Conditions in Compact Spaces

DMS-0103985 Alan DowThe Principal Investigator建议专注于集合论拓扑中的三个中心问题，涉及基本的可数收敛问题和集合论独立性结果。第一个问题，由于Efimov，是要确定是否是一致的，任何不包含收敛序列的紧空间将包含整数的Stone-Cech紧化。第二，由于Bashkirov，是确定是否可以有一个可数界上的序列顺序的一个紧凑的序列空间。在某些方面，这个问题可以被看作是最近解决的摩尔-姆洛卡问题的一个非常有趣的加强。第三个是继续系统研究的斯通-切赫其余的实数类似的，这已经进行了很长时间的其余的整数。似乎有许多障碍所造成的额外的复杂性所施加的连通性property.General收敛序列的存在性和收敛序列在多大程度上完全决定了拓扑结构的问题经常出现在各种设置。这些问题在函数空间的紧致子集（实值函数集上的naturalinformative拓扑）的背景下特别有意义。调查人员正在继续追求的调查线，刺激了密集的相互作用领域之间的一般拓扑学和集合论的发展。

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