Applications of Set Theory to Topology and Topology to Model Theory

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This research applies one field of mathematics (Set Theory) to another (Topology), as well as the second to a third (Model Theory). Set Theory is the study of the fundamental building blocks (sets) of the mathematical universe; Topology is the study of abstractions of geometric properties of space; Model Theory is the study of languages and how languages relate to the structures they describe. By definition, Pure Mathematics, such as these fields, does not concern itself with real-world applications. Nonetheless, time after time, such basic research has led to important technological advances, such as the development of the computer, which were not foreseen by the early pioneers in such research. Therefore, wise governments fund basic research. Applications of Set Theory to Topology, which were pioneered by the author and a few colleagues almost 50 years ago, have revolutionized the more abstract portions of the latter field by showing its dependence on underlying Set Theory, with the possibility of producing both theorems and counterexamples for the same question, depending on what set-theoretic principles beyond the usual ones used in Mathematics one adopts, because those usual principles do not settle such questions either way. Recently, we have been using new set-theoretic methods to settle longstanding open questions in Topology, and our proposed program continues along these lines. Consumers of such work are topologists and/or set-theorists. Applications of Topology to Model Theory have a long history, but the investigator noticed that such applications used little beyond what could be found in elementary topology textbooks. He has therefore championed the use of more sophisticated topological methods in Model Theory, and was vindicated when his doctoral student, C. Eagle, produced an important thesis using such methods. The proposed research program will see Eagle and the investigator continue these topological applications. The Baire Category Theorem is widely used in many areas of Mathematics; it allows one to assert the satisfaction of infinitely many requirements on an object one is constructing. Topologists have greatly extended the kinds of situations in which this theorem can be applied, but Model Theorists have until now not made use of these extensions. Eagle and the investigator will be using these extensions to make progress on fundamental questions in Model Theory and its applications, concerning what properties can be described in which languages, so that researchers in e.g. Analysis (the field of Mathematics that deals with limits, extending Calculus), can make use of them.