Mikhail Yakovlevich Souslin.

Descriptive set theory, Fall 2007

Instructor: Christian Rosendal, room 304 Altgeld Hall

Course hours: 10:00 - 10:50 AM, MWF.
Location: Room 441, Altgeld Hall.

Homework sets:
  • Homework I pdf

  • Homework II pdf

  • This is a course covering the core material of classical descriptive set theory. Descriptive set theory concerns the structure and regularitry properties of definable subsets of Polish spaces, e.g., definable subsets of the reals. It is well-known that using the axiom of choice bad sets of reals can be constructed, e.g., non-measurable sets, but if one only consider sets that are defined explicitly provably such sets do not occur. Descriptive set theory is the study of these explicitly defined sets. Descriptive set theory thrives in its interactions with other branches of mathematics such as the study of inner models of set theory, the geometry of Banach spaces, ergodic theory, and harmonic analysis and has proved to be a useful tool in all of these domains. So the course will be of interest to the general abstract analyst. Among the topics we will cover are:

  • Borel, analytic, and coanalytic sets.
  • Baire category.
  • Baire class 1 functions.
  • Separation theorems.
  • Uniformisation theorems.
  • Infinite games and determinacy.
  • Coanalytic ranks and scales.

  • Much of the material will presuppose a certain maturity in analysis that can be gained from courses on measure theory, functional analysis, or general topology. A knowledge of set theory is also useful but not essential. There are no specific required prerequisites.

    Required reading:
  • Alexander S. Kechris: Classical descriptive set theory, Graduate texts in Mathematics 156, Springer Verlag 1995.

  • Other suggested reading:
  • S. M. Srivastava: A course on Borel sets, Springer 1998.
  • Y. Moschovakis: Descriptive set theory, North Holland 1980.
  • Galen & Weitkamp: Recursive aspects of descriptive set theory, Oxford 1985.
  • K. Kuratowski: Topology, Vol. 1.

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