Math 873: Descriptive Set Theory
It is straightforward to use the Axiom of Choice to produce a pathological subset of the real line, for example, one that is not Lebesgue measurable, or one that is uncountable but contains no nonempty perfect subsets. On the other hand, sufficiently nice subsets of the real line do not exhibit these pathologies. Borel sets and even $\Sigma^1_1$ sets (i.e., continuous images of Borel sets) are measurable and have the perfect set property. Descriptive Set Theory is the study of such classes of “well-behaved” sets. We will give an introduction to classical Descriptive Set Theory on Polish spaces, including the Borel Hierarchy, the determinacy of Borel games (a celebrated theorem of Martin), and regularity properties of $\Sigma^1_1$ sets. Time permitting, we will discuss the complexity of Borel equivalence relations.
UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, WI 53706