Borel extractions of converging sequences in compact sets of Borel functions

Ideals of compact sets associated with Borel functions

We investigate the connection between the Borel class of a function f and the Borel complexity of the set C(f) = {C ∈ J(X) : f |C is continuous} where J(X) denotes the compact subsets of X with the Hausdorff metric. For example, we show that for a function f : X → Y between Polish spaces; if C(f) is Fσδ in J(X), then f is Borel class one.

Towers of Borel Functions

We give mathematical reformulations of the cardinals p and t in terms of families of Borel functions. As an application we show that t is invariant under the addition of a single Cohen real.

Disjoint Borel functions

For each a ∈ ω, we define a Baire class one function fa : ω → ω which encodes a in a certain sense. We show that for each Borel g : ω → ω, fa ∩ g = ∅ implies a ∈ ∆1(c) where c is any code for g. We generalize this theorem for g in a larger pointclass Γ. Specifically, when Γ = ∆2, a ∈ L[c]. Also for all n ∈ ω, when Γ = ∆3+n, a ∈M1+n(c).

A Course on Borel Sets

The hierarchy of ω 1 - Borel sets 1 The hierarchy of ω 1 - Borel sets Arnold

We consider the ω1-Borel subsets of the reals in models of ZFC. This is the smallest family of sets containing the open subsets of the 2 and closed under ω1 intersections and ω1 unions. We show that Martin’s Axiom implies that the hierarchy of ω1-Borel sets has length ω2. We prove that in the Cohen real model the length of this hierarchy is at least ω1 but no more than ω1 + 1. Some authors have...