Set-theoretic methods and the study of compact spaces

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This project investigates fundamental and foundational questions in connection with convergence or limiting properties of infinite sets in an ambient space. The issues of the long-term behavior of an infinite sequence of processes or systems (however abstract) is at the core of many mathematical or physical questions. The universal setting for this investigation is known as the study of compact topological spaces. The "points" in the ambient space can represent any mathematical object in applications ranging from turbulent dynamical systems to a quantum mechanics setting. It has emerged that many natural questions are sensitive to the very foundational principles of mathematics itself. This research project pursues the resolution of such topological questions armed with the modern methods of uncovering possible dependence on the foundational axioms of mathematics. The investigator will study the influence of well-known countability properties on the structure of compact (Hausdorff) spaces. Some spaces are determined by the behavior of their converging sequences, while one of the most fundamental spaces, henceforth BetaN, the Stone-Cech compactification of the integers N, are at the opposite extreme in that it has no converging sequences whatsoever. The behavior of BetaN is heavily influenced by extra axioms of set-theory and many natural questions about spaces of functions and measures and the like, are as well. Additionally, the Banach spaces of continuous real-valued functions with domain some compact subset of BetaN will be investigated. Again set-theoretic hypotheses, mixed with additional topological properties, lead to varying behavior in the important class of compact spaces. Our study is the analysis of the interplay between the various countable convergence conditions and the axioms of set-theory that influence them. We are expecting to have to develop new techniques of constructing topological spaces as well as the refinement of forcing techniques to uncover basic combinatorial structures involved. These specific problems of longstanding will be of central focus. (1) If an infinite compact space has no converging sequence, does it contain a copy of BetaN? (2) Are the metrizable compact spaces the only ones whose diagonal is uncountably-inaccessible? (3) Is the Banach space of continuous functions on the remainder BetaN - N, like BetaN - N itself, essentially unique in certain models of set theory? (4) If the topology of a compact space is determined by converging countable sequences, is there a finite or countable bound on how many times one must iterate taking limits to capture all such limits?