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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

  1. Srivastava, A course in Borel sets
  2. Kechris, Descriptive set theory
  3. Jech, Set theory

Homework 1 (deadline 25.04)

  1. 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

  1. Metric and topological spaces
    1. Metric spaces
    2. Product, topological sum
    3. Compact and sequentially compact spaces
    4. Comparison of compact and sequentially compact spaces in general topological context
    5. Cech compactification of $\omega$, i.e. the space of ultrafilters on $\omega$
    6. Space $\{0,1\}^{[0,1]}$
  2. Ordinals and cardinals
    1. Transitive sets
    2. Axiom of Regularity
    3. Axiom of Infinity
    4. Some examples of ordinals: $0,1,2,\omega,\omega+\omega+3,\omega_1$
    5. Connection to well orderings and Axiom of Choice
  3. Polish spaces
    1. Compact and sequentially compact spaces in the class of metric spaces
    2. Compact metrizable spaces are Polish
    3. $\mathbb{R}$ $\sigma$-compact poishh space
    4. $\omega^\omega$ polish space which is not $\sigma$-compact
  4. Borel sets
    1. Algebras and $\sigma$-algebras of sets
    2. Algebra, $\sigma$-algebra generated by some family
    3. How to build algebra $\sigma$-algebra generated by $\mathcal{A}$
    4. Borel hierarchy
  5. Universal sets
    1. Universal open set
    2. Universal closed set
    3. Universal $F_\sigma$-set
    4. General case: universal $\Sigma^0_{\alpha}$ and $\Pi^0_{\alpha}$ sets
    5. Baire space as a $G_{\delta}$ subspace of uncountable Polish space
    6. Universal sets as subsets of $X\times X$
    7. $\Delta^0_{\alpha}$ classes
    8. There is no universal set for $\Delta^0_{\alpha}$, Borel
  6. Borel codes
  7. More on Borel classes
    1. Reduction theorem for $\Sigma^0_{\alpha}$
    2. Separation theorem for $\Pi^0_{\alpha}$
    3. Failure of separation for $\Sigma^0_{\alpha}$
  8. Analytic, coanalytic sets
    1. Different characterizations of analytic sets
    2. Universal analytic set
    3. Analytic sets are closed under countable unions and intersections but not under complements
    4. There is analytic set which is not Borel
    5. Separation theorem for analytic sets
    6. $Borel=\Sigma^1_1\cap\Pi^1_1$

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