MATH 7870 - Set Theory

Justin Moore, spring 2017.

This will be an introductory course on descriptive set theory. Classically descriptive set theory is the study of families of regular subsets of Euclidean space and other Polish spaces (separable, completely metrizable spaces). The most important families of sets exhibiting regularity properties are the Borel sets and their continuous images, known as the analytic sets. The course will begin with a classical analysis of Polish spaces, the hierarchy of their Borel sets their analytic and co-analytic sets. This will include the separation theorem for analytic sets, Hurewicz's dichotomy, uniformization theorems, the boundedness theorem and perfect set property of analytic sets. The second part of the course will cover more modern topics: the basic theory of Borel equivalence relations, the determinacy of Gale-Stewart games, and the Borel chromatic number Borel graphs.

Prerequisites

Students are expected to have basic familiarity with real analysis, the topology of metric spaces, and measure theory. Background in set theory and logic is not required.

Optional Textbook

Classical Descriptive Set Theory, by Alexander Kechris (Springer Graduate Texts in Mathematics)