Selected topics in set theory 1000-1M09WZM
This advanced course will be devoted to selected topics of Descriptive Set Theory. It is a branch of Set Theory which studies "definable" subsets of the reals (and similar spaces like R^n, the Cantor set and other spaces of infnite sequences or trees). "Definable" sets include, in particular, Borel sets and their continuous images. Such sets have many regular properties: they are Lebesgue measurable, have the Baire property (which is a topological analogue of measurability) and the Continuum Hypothesis restricted to their class is valid. The notions and results from Descriptive Set Theory have applications in variousbranches of Mathematics and also in Theoretic Computer Science.
During the course we will present some partition theorems, i.e., descriptive counterparts of the Ramsey theorem such as the Galvin-Prikry, Silver and Ellentuck theorems. We will show some uniformization results concerning Borel and projective sets. We will consider questions concerning Borel games and their determinacy.
We are also planning to present connections between Descriptive Set Theory and Automata Theory, more precisely: applications of Descriptive Set Theory to problems concerning sets of infinite sequences or trees acceptable by finite automata.
We assume familiarity with basic Set Theory (including the transfinite induction, ordinal and cardinal numbers) and Topology.
Type of course
Bibliography
1. A.S. Kechris, Classical descriptive set theory, Graduate Texts in Math. 156, Springer-Verlag, 1995.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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