## Math 223DWinter 2008Descriptive Set Theory

Room and Time

MWF 11:00-11:50 in MS 5217

Note: If it is convenient for the students, I would like to change to a two-day schedule.

Instructor

• D.A. Martin

• Offices: MS 7935 and Dodd 355

• Office Hours TBA

• Email Address: dam@math.ucla.edu

Content

Descriptive set theory is usually characterized as the study of definable sets in Polish spaces (separable, completely metrizable spaces). The subject classifies definable sets into hierarchies based on the complexity of their definitions. For each hierarchy, one studies such things as the properties shared by all sets in the hierarchy and the structural properties of the levels of the hierarchy. Examples of hierarchies are the Borel hierarchy, the projective hierarchy, and the Wadge hierarchy, the last of these being an ultra-fine hierarchy that refines and extends most or all the others. An example of a property shared by all sets in a hierarchy is universal measurability, which is a property of all Borel sets (i.e., of all sets in the Borel hierarchy).

Text

There is no text for the course, but Alexander S. Kechris's book, Classical Descriptive Set Theory, published by Springer-Verlag, contains all material the course will cover. Moreover the course will have almost the same prerequisites as the book (but fewer) and and will use pretty much the same terminology and notation. The course will also follow the book in dealing (almost exclusively) with classical descriptive set theory as opposed to effective descriptive set theory, which refines the classical theory using the concept of recursive (computable) function. Finally, the course will follow the book in making considerable use of infinite games.

Prerequisites

What's needed is familiarity with elementary set theory, including the notions of wellordering, ordinal number, and transfinite induction, and a little acquaintance with the basics of general topology and measure theory.

Course work

One or two homework problems will be assigned every week or two.