Representing pointsets as unions of Borel sets

Representing Projective Sets as Unions of Borel Sets

We consider a method of representing projective sets by a particular type of union of Borel sets, assuming AD. We prove a generalization of the theorem that a set is S^ iff it is the union of a>x Borel sets. The Axiom of Determinacy (AD) is always assumed. Theorem 1 (Sierpinski, Solovay, Moschovakis [8, 7D.10]). Let A c com. A is Ej iff A is the union of cox Borel sets. The purpose of this pape...

Finite Sets as Complements of Finite Unions of Convex Sets

Suppose S ⊆ R is a set of (finite) cardinality n whose complement can be written as the union of k convex sets. It is perhaps intuitively appealing that when n is large k must also be large. This is true, as is shown here. First the case in which the convex sets must also be open is considered, and in this case a family of examples yields an upper bound, while a simple application of a theorem ...

Descriptive complexity of countable unions of Borel rectangles

We give, for each countable ordinal ξ ≥ 1, an example of a ∆2 countable union of Borel rectangles that cannot be decomposed into countably many Πξ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ∆ξ-measurable countable coloring. 2010 Mathematics Subject Classification. Primary: 03E15, Secon...

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...