Mathematical Sciences: Problems in Set-Theoretic Topology

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Set-theoretic topology is a branch of topology in which set- theoretic techniques are used to solve problems about abstract spaces. In it, special axioms (such as Godel's Axiom of Constructibility, Martin's Axiom, the covering Lemma, and the Product Measure Extension Axiom), forcing techniques, and other set-theoretic tools are used to clarify, and often solve, long- outstanding problems, and through consistency results to isolate and identify topological properties worthy of study. The principal investigator will continue his research in this branch, centered on four areas where recent results have greatly enhanced our understanding of the spaces in question and, in some cases, the set-theoretic tools themselves. The spaces in question are various classes of countably compact spaces; Frechet chain net spaces; nonmetrizable manifolds; and countably metacompact spaces. Such studies will probably have less immediate application than more geometric topological studies but in the long run can be expected to influence our ideas about space and even about the nature of proof.