NSCPHD1220 Descriptive set theory (DesSet)

Volume 2014/2015
Content

Polish spaces and standard Borel spaces. The Borel hierarchy. Analytic sets and tree representations. Lusin's separation theorem. Baire and Lebesgue measurability, Kuratowki-Ulam theorem and other regularity properties. Selection theorems for Borel relations. Borel and analytic equivalence relations. Polish groups and their actions; orbit equivalence relations. Borel reducibility, and the dichotomy theorems of Silver and Harrington-Kechris-Louveau, and possibly other topics.

Learning Outcome

Knowledge: The student should know the definitions of Polish spaces, standard Borel spaces, and examples of these, as well as the definition of the Borel hierarchy, of analytic sets, and for their tree analysis; Lusin's separation theorem and its consequences, and the regularity properties of analytic sets; the selection problem for Borel relations, as well as the Jankov-von Neumann selection theorem, and the selection principle for Borel relations with countable sections; the concenpt of genericity together with the Kuratowski-Ulam theorem; the concept of Borel reducibility, and the basic dichotomies of Silver and Harrington-Kechris-Louveau.

Skills: The student should be able to apply descriptive set theoretic concepts and result mentioned in the previous paragraph to prove borelness/analyticity of a relation/function, check whether a given set is generic/meager, apply basic dichotomies to equivalence relations and solve other problems related to the material of the course.

Competences:  The student should be able to formulate the main results of the course, check whether they are applicable in a concrete problem and use them to solve it.

General topology and measure theory.
4 hours of lectures/week + 2 hours of exercises per week for 9 weeks.
  • Category
  • Hours
  • Exercises
  • 14
  • Lectures
  • 28
  • Preparation
  • 164
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Continuing evaluation based on three problem sets graded on the 7-point scale.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Re-exam
30 min oral examination, no preparation time.
Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.