Back
Descriptive set theory (2023/2024)
Lectures take place on Thursday 15:15 - 17:00 in room A.1.2 C-19.
This course is for PhD students.
Materials and literature
- Srivastava, A course in Borel sets
- Kechris, Descriptive set theory
- Jech, Set theory
Homework 1 (deadline 25.04)
- Prove that every uncountable Polish space $X$ has size continuum. Find a (continuous) injection $f:2^\omega\to X$.
Final mark
There will be some homeworks during the semester.
In the end there will be oral exam.
Schedule
- Metric and topological spaces
- Metric spaces
- Product, topological sum
- Compact and sequentially compact spaces
- Comparison of compact and sequentially compact spaces in general topological context
- Cech compactification of $\omega$, i.e. the space of ultrafilters on $\omega$
- Space $\{0,1\}^{[0,1]}$
- Ordinals and cardinals
- Transitive sets
- Axiom of Regularity
- Axiom of Infinity
- Some examples of ordinals: $0,1,2,\omega,\omega+\omega+3,\omega_1$
- Connection to well orderings and Axiom of Choice
- Polish spaces
- Compact and sequentially compact spaces in the class of metric spaces
- Compact metrizable spaces are Polish
- $\mathbb{R}$ $\sigma$-compact poishh space
- $\omega^\omega$ polish space which is not $\sigma$-compact
- Borel sets
- Algebras and $\sigma$-algebras of sets
- Algebra, $\sigma$-algebra generated by some family
- How to build algebra $\sigma$-algebra generated by $\mathcal{A}$
- Borel hierarchy
- Universal sets
- Universal open set
- Universal closed set
- Universal $F_\sigma$-set
- General case: universal $\Sigma^0_{\alpha}$ and $\Pi^0_{\alpha}$ sets
- Baire space as a $G_{\delta}$ subspace of uncountable Polish space
- Universal sets as subsets of $X\times X$
- $\Delta^0_{\alpha}$ classes
- There is no universal set for $\Delta^0_{\alpha}$, Borel
- Borel codes
- More on Borel classes
- Reduction theorem for $\Sigma^0_{\alpha}$
- Separation theorem for $\Pi^0_{\alpha}$
- Failure of separation for $\Sigma^0_{\alpha}$
- Analytic, coanalytic sets
- Different characterizations of analytic sets
- Universal analytic set
- Analytic sets are closed under countable unions and intersections but not under complements
- There is analytic set which is not Borel
- Separation theorem for analytic sets
- $Borel=\Sigma^1_1\cap\Pi^1_1$
Back