Print syllabus(in Polish) Ideały miary i kategorii
General data
| Course ID: | 1000-1M23ITM |
| Erasmus code / ISCED: |
11.1
|
| Course title: | (unknown) |
| Name in Polish: | Ideały miary i kategorii |
| Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
| Course groups: |
Elective courses for 2nd stage studies in Mathematics |
| ECTS credit allocation (and other scores): |
6.00
|
| Language: | English |
| Type of course: | elective monographs |
| Requirements: | Set theory 1000-135TMN |
| Prerequisites (description): | Completing the courses of Set Theory and Topology is required, as well as the knowledge of basics of measure theory from Mathematical Analysis II. |
| Short description: |
The lecture will be devoted to set-theoretic properties of ideals of Lebesgue measure zero sets and Baire first category sets, as well as other related classes of small subsets of the real line. |
| Full description: |
1. Real line and related Polish spaces, the Cantor and Baire spaces. Elements of descriptive set theory: borel and analytic sets. Perfect sets and the perfect set property for classes of subsets of Polish spaces. 2. Ideals of measure and category as c.c.c. ideals with Borel bases. Quotient algebras of Borel sets modulo an ideal, Sikorski's theorem. Baire property as a "category" version of measurability. The Kuratowski-Ulam Theorem as a counterpart of the Fubini Theorem. Orthogonality of ideals of measure and category. Erdős-Sierpiński duality theorem (under CH), non-existence of an additive Erdős-Sierpiński mapping. 3. Theorems and constructions regarding non-measurable sets or sets w/o the Baire property, e.g. The Four Poles Theorem. Nonmeasurable algebraic sums of sets from the ideal. 4. Cardinal characteristics of the ideals of measure and category and the inequalities between them (e.g. Rothberger's inequality). Cichoń's diagram. 5, Universal Measure Zero sets and their catgeory counterparts: Always of First Category Sets and Universally of First Category Sets. and their properties, e.g. the existence of uncountable sets with these properties. 6. Strong Measure Zero sets, their metric definition and the characterization by Galvin-Mycielski-Solovay. Their category counterparts - strongly meager and very meager sets. Sets with Rothberger's property and other related classes. Luzin and Sierpiński sets. Information about the Borel Conjecture and the Dual Borel Conjecture. |
| Bibliography: |
Research papers and selected topics from: J. C. Oxtoby - Measure and Category (2nd Edition), Springer Verlag. Alexander S. Kechris - Classical Descriptive Set Theory, Springer Verlag. T. Bartoszyński, H. Judah - Set Theory. On the structure of the real line. A.K. Peters Ltd. |
| Assessment methods and assessment criteria: |
Oral exam at the end of the course. Student's activity on exercise classes may influence the final grade. |
Classes in period "Summer semester 2023/24" (in progress)
| Time span: | 2024-02-19 - 2024-06-16 |
 
MO TU W TH FR WYK
CW
|
| Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
| Coordinators: | Marcin Kysiak | |
| Group instructors: | Marcin Kysiak | |
| Students list: | (inaccessible to you) | |
| Examination: | Examination |
Copyright by University of Warsaw.
