At a certain basic level topology merges into set theory, which makes it a part of the foundations of mathematics. The principal investigator will continue his research on several open questions in set-theoretic topology. One theme is normality and countable paracompactness versus paracompactness in locally compact spaces. Several longstanding problems in this area remain unsettled, e.g., the Arhangel'skii-Tall problem (Are normal locally compact metacompact spaces paracompact?) and Watson's problem (Is there a perfectly normal locally compact non- paracompact space in ZFC?) Another problem has led to interesting combinatorics involving ladder systems that will likely have further applications. Other problems to be investigated include the existence of Dowker filters, and questions of Watson and Cook involving connectedness. Possible methods of solution include set- theoretic axioms and combinatorics, and forcing. Solutions to the problems would deepen our understanding of fundamental topological properties and constructions, and would likely require new techniques applicable to a variety of other problems.
