Descriptive set theory

Syllabus

In classical descriptive set theory, we are interested in sets appearing naturally in various topics in mathematics, in particular in functional analysis, harmonic analysis, dynamical systems and group theory. One of the goals is to study the topological complexity of these sets. For example, one can classify the Borel subsets of the real line by considering the number of steps necessary to construct them, starting from the open sets, and allowing countable unions and taking complements.

We work in Polish topological spaces, where Baire’s theorem is a powerful tool. We will first be interested in the Borel subsets of the Polish spaces, and we will see that the countable ordinals define a natural hierarchy among them. Then we will study the direct images of the Borel sets by the Borel maps (the analytic sets) and their complement (the co-analytic sets). In particular, we will provide a method allowing us to prove that some sets are co-analytic but not Borel.

We will finish this course with an introduction to effective descriptive set theory and its appli- cations. One of its very powerful tool is the Gandy-Harrington topology, and we will establish its properties allowing to use it to prove many dichotomy results. We will provide the details for at least three examples: the Hurewicz dichotomy, the Silver dichotomy and the Kechris-Solecki-Todorčević dichotomy. We will state some more recent examples, and give some details if time permits.

Contents

• General topology
• Polish spaces
• The Cantor and the Baire spaces - Baire category
• Borel sets and Borel functions
• Analytic sets and co-analytic sets - Effective descriptive set theory
• Application to dichotomies

Bibliography

• S. Gao, Invariant Descriptive Set Theory, Pure and Applied Mathematics, A Series of Mono- graphs and Textbooks, 293, Taylor and Francis Group, 2009
• A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995
• A. S. Kechris, S. Solecki and S. Todorčević, Borel chromatic numbers, Adv. Math. 141 (1999), 1-44 [M] Y. N. Moschovakis, Descriptive set theory, North-Holland, 1980

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