Print syllabus(in Polish) Wybrane zagadnienia teorii mnogości małych podzbiorów przestrzeni polskich
General data
| Course ID: | 1000-1M21TMPP |
| Erasmus code / ISCED: |
11.1
|
| Course title: | (unknown) |
| Name in Polish: | Wybrane zagadnienia teorii mnogości małych podzbiorów przestrzeni polskich |
| Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
| Course groups: |
(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics |
| ECTS credit allocation (and other scores): |
(not available)
|
| Language: | English |
| Type of course: | elective monographs |
| Prerequisites (description): | To attend the course, it is necessary to have knowledge of set theory somewhat beyond the content of a standard introductory course (including transfinite induction, ordinal and cardinal numbers) and elementary notions of general topology on the level of "Topology I" course. |
| Short description: |
The course will deal with some topics lying on the borders of descriptive set theory, general topology, and measure theory. They are connected to ideals on countable sets and σ-ideals on Polish spaces, i.e., separable, completely metrizable topological spaces. The notion of an ideal on an infinite set X corresponds to the intuition of a family of subsets of this set which are small from a certain point of view. So it is a family of subsets of X, different from P(X), closed under subsets and finite unions if its elements; it is a σ-ideal if it is moreover closed under countable unions. Ideals are present in many parts of mathematics and various problems, concerning them, are the subject of interesting and current research, carried out by many well-known mathematicians. |
| Full description: |
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. |
| 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. |
| 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 methods and assessment criteria: |
An exam |
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