Applications of Set Theory to Topology and Topology to Model Theory

本研究将一个数学领域（集合论）应用于另一个数学领域（拓扑学），以及第二个数学领域应用于第三个数学领域（模型论）。集合论是对数学宇宙的基本构建块（集合）的研究;拓扑学是对空间几何属性的抽象的研究;模型论是对语言以及语言如何与它们所描述的结构相关联的研究。根据定义，纯数学，如这些领域，不关心自己与现实世界的应用。尽管如此，一次又一次，这种基础研究导致了重要的技术进步，如计算机的发展，这是早期的先驱者没有预见到的。因此，明智的政府资助基础研究。* 应用集理论拓扑，这是由作者和几个同事开创了近50年前，已经彻底改变了后者领域的更抽象的部分，通过显示其对基础集理论的依赖，有可能产生定理和反例为同一个问题，这取决于什么集理论原则超出了数学中常用的原则，因为这些通常的原则无论如何都不能解决这些问题。最近，我们一直在使用新的集合理论的方法来解决长期悬而未决的问题，拓扑，我们提出的计划继续沿着沿着这些路线。这种工作的消费者是拓扑学家和/或集合理论家。* 拓扑在模型论中的应用有着悠久的历史，但研究人员注意到，这些应用几乎没有超出初等拓扑教科书中的内容。因此，他支持在模型论中使用更复杂的拓扑方法，当他的博士生C。鹰，产生了一个重要的论文使用这种方法。拟议的研究计划将看到鹰和研究人员继续这些拓扑应用。Baire范畴定理被广泛应用于数学的许多领域;它允许人们断言对一个正在构造的对象的无限多个要求的满足。拓扑学家已经极大地扩展了这一定理可以应用的情形，但是模型理论家直到现在还没有利用这些扩展。Eagle和研究者将使用这些扩展在模型论及其应用的基本问题上取得进展，关于什么性质可以用什么语言描述，以便分析（处理极限的数学领域，扩展微积分）的研究人员可以使用它们。

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.**