A reflection phenomenon in descriptive set theory

Descriptive Set Theory Descriptive Set Theory

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

Classical descriptive set theory as a refinement of effective descriptive set theory

The (effective) Suslin-Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability interpretation. Effective Descriptive Set Theory is advertized as a refinement of the classical theory of definability (on Polish spaces) developed in the first half of the 20th century, for exam...