Topics in invariant descriptive set theory

Descriptive Set Theory Descriptive Set Theory

Topics in Set Theory

Axiomatics. The formal axiomatic system of ordinary set theory (ZFC). Models of set theory. Absoluteness. Simple independence results. Transfinite recursion. Ranks. Reflection principles. Constructibility. [4] Infinitary combinatorics. Cofinality. Stationary sets. Fodor’s lemma. Solovay’s theorem. Cardinal exponentiation. Beth and Gimel functions. Generalized Continuum Hypothesis. Singular Card...

Descriptive Set Theory Problem Set

Prove that any strictly monotone sequence (Uα)α<γ of open subsets of X has countable length, i.e. γ is countable. Hint: Use the same idea as in the proof of (a). (c) Show that every monotone sequence (Uα)α<ω1 open subsets of X eventually stabilizes, i.e. there is γ < ω1 such that for all α < ω1 with α ≥ γ, we have Uα = Uγ. Hint: Use the regularity of ω1. (d) Conclude that parts (a), (b) and (c)...

Some descriptive set theory

Definition 1.1. Let (X, τ) be a topological space. A subset D ⊆ X is called dense if D ∩O 6= ∅ for every nonempty open set O ⊆ X. X is called separable if X has a countable dense subset. X is called metrizable if there is a metric d on X such that the topology τ is the same as the topology induced by the metric. The metric is called complete if every Cauchy sequence converges in X. Finally, X i...

Descriptive Set Theory