Mathematical Scieces: Topics in Set-Theoretic Topology

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9401529 Gruenhage The principal investigator will continue his research on several open questions in set-theoretic topology. The focus will be on the now classical problem of Arhangel'skii and Tall: Are normal locally compact metacompact spaces paracompact? Because of recent progress by the principal investigator, a solution to this problem may be near. Another set of problems revolves around the failure of the Lindelof property in products, and includes Michael's long-standing question concerning the existence in ZFC of of a Lindelof space whose product with a complete separable metric space is not Lindelof. Yet a third set of problems involves extensions of the PI's recent solutions to problems of Watson, and Dow and Porter, on closed partitions of compact Hausdorff spaces. Set-theory and general topology are fundamental mathematical disciplines which have common turn-of-the-century roots. Set- theory provides a framework upon which all of mathematics can be based. Objects of study in topology include familiar spaces such as the plane and ordinary three-dimensional Euclidean space, and their subsets, as well as much more abstract spaces. Pioneering work of Goedel and Cohen led to the realization that some long- standing topological questions could not be solved assuming only the standard axioms of set-theory. It is illuminating to determine whether or not topological propositions are independent of the standard axioms in this sense, and if so, what further axioms are necessary to settle them. This project involves just such questions, concerning basic topological properties and constructions, including product spaces, partitions, and refinements of coverings. Questions to be studied include ones which date back to the 1960's and have been worked on by many experts with only partial success so far. Significant recent progress by the principal investigator indicates that a complete solution to some of these may now be at hand. The new tech niques involved in their solution are likely to be applicable to a wide variety of other problems. ***