Study of Countable Convergence Conditions in Compact Spaces

University Of North Carolina At Charlotte, Charlotte NC

Investigators

Abstract

DMS-0103985 Alan Dow The 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 (natural informative 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