Topics in Set-theoretic Topology

Auburn University, Auburn AL

Investigators

Abstract

The principle investigator will explore open problems in several areas of set-theoretic topology. Convergence properties have important implications in a number of fields, yet are not well-understood. For example, it is not known if it is consistent that a space which is Frechet-Urysohn for finite sets must be first-countable. Settling this question is likely to lead to a solution of Malychin's well-known problem whether it is consistent that separable Frechet-Urysohn topological groups must be metrizable. Two other themes of this proposal involve function spaces with the compact-open topology. Settling one of them would solve a 40 year old problem of J. Ceder, and settling the other would yield a useful way of determining when these function spaces are Baire spaces. Another goal of this proposal is to apply Balogh's magnificent technique for constructing ZFC examples , which he used to settle several long-standing conjectures, to other problems, and in the process attempt to 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, a property with interesting but not well-understood relationships with covering properties, which indicate that the time may be ripe for settling the D-space questions of van Douwen that have stumped researchers for more than 20 years. Set theory and general topology are fundamental mathematical disciplines, with common historical roots, and they serve as essential tools in many areas of mathematics. General topology provides a framework for the study of shapes, from ordinary shapes in real three-dimensional space to much more abstract shapes and structures. For example, continuous real-valued functions are a staple of mathematics, and hence the topological structure of spaces of continuous functions, which is one of the topics that this proposal centers on, has important implications in many fields. The same can be said about the notion of sequential convergence and its variants, another topic of this proposal. The principle investigator's proposed problems lie within the scope of what has been a fruitful interaction between general topology and set theory, an interaction spurred by dramatic advances in set theory and logic in the last forty years or so and the realization that many long-standing questions in the general topology of abstract spaces rest on complicated set-theoretic combinatorics. Solutions to the proposed problems would deepen our understanding of the fundamental topological properties to be explored, and would likely require new set-theoretical and topological techniques applicable to a variety of other problems.