Topics in Set-theoretic Topology

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The principle investigator will explore open problems in several areas ofset-theoretic topology. Convergence properties have importantimplications in a number of fields, yet are not well-understood. Forexample, it is not known if it is consistent that a space which isFrechet-Urysohn for finite sets must be first-countable. Settling thisquestion is likely to lead to a solution of Malychin's well-known problemwhether it is consistent that separable Frechet-Urysohn topological groupsmust be metrizable. Two other themes of this proposal involve functionspaces with the compact-open topology. Settling one of them would solve a40 year old problem of J. Ceder, and settling the other would yield auseful way of determining when these function spaces are Baire spaces.Another goal of this proposal is to apply Balogh's magnificent techniquefor constructing ZFC examples , which he used to settle severallong-standing conjectures, to other problems, and in the process attemptto simplify and codify the technique to make it easier to use. Finally,there have been a number of recent advances on the D-space property, aproperty with interesting but not well-understood relationships withcovering properties, which indicate that the time may be ripe for settlingthe D-space questions of van Douwen that have stumped researchers for morethan 20 years. Set theory and general topology are fundamental mathematical disciplines,with common historical roots, and they serve as essential tools in manyareas of mathematics. General topology provides a framework for the studyof shapes, from ordinary shapes in real three-dimensional space to muchmore abstract shapes and structures. For example, continuous real-valuedfunctions are a staple of mathematics, and hence the topological structureof spaces of continuous functions, which is one of the topics that thisproposal centers on, has important implications in many fields. Thesame can be said about the notion of sequential convergence and itsvariants, another topic of this proposal. The principle investigator'sproposed problems lie within the scope of what has been a fruitfulinteraction between general topology and set theory, an interactionspurred by dramatic advances in set theory and logic in the last fortyyears or so and the realization that many long-standing questions in thegeneral topology of abstract spaces rest on complicated set-theoreticcombinatorics. Solutions to the proposed problems would deepen ourunderstanding of the fundamental topological properties to be explored,and would likely require new set-theoretical and topological techniquesapplicable to a variety of other problems.