(in Polish) Wybrane zagadnienia teorii mnogości małych podzbiorów przestrzeni polskich 1000-1M21TMPP
1. Elements of descriptive set theory: Polish spaces, Borel and analytic sets, the hyperspace of compact sets, the Baire Property, the σ-ideal of sets of the first Baire’a category and its characterizations.
2. Ideals on countable sets: Talagrand's characterization of ideals with the Baire Property, Mazur's characterization of F_σ ideals and Solecki's characterization of analytic P-ideals with the help of submeasures.
3. σ-ideals generated by closed sets in Polish spaces: a construction of G_δ-sets not in the σ-ideal (Hurewicz systems and Solecki's theorem), σ-ideals with the "1-1 or constant" property of Sabok and Zapletal (every Borel function from a Borel set not in the ideal into a Polish space is 1-1 or constant on a Borel set not in the ideal).
4. σ-ideals of small sets in the sense of measure or category: universal measure zero sets, strong measure zero sets, perfectly meager sets, universally meager sets, strongly meager sets.
Type of course
Prerequisites (description)
Learning outcomes
The student:
1. is familiar with the basics of descriptive set theory, including classical examples of Polish spaces, definitions of Borel and analytic sets, sets with the Baire property and first category sets,
2. knows how to use some known characterizations of these ideals on countable sets which have the Baire property or are of type F_σ or analytic P-ideals,
3. is familiar with some special properties of σ-ideals generated by closed sets in Polish spaces,
4. can show examples of sets that are small in the sense of measure or category and can describe how various classes of such sets are related.
Assessment criteria
An exam
Bibliography
[1] A. S. Kechris, Classical descriptive set theory, Graduate Texts in Math. 156, Springer-Verlag (1995).
[2] S. Solecki, Covering analytic sets by families of closed sets, Journal of Symbolic Logic 59(3) (1994), 1022–1031.
[3] S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), 51–72.
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:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: