On the decomposition of sets of reals to borel sets

Borel-amenable reducibilities for sets of reals

We show that if F is any “well-behaved” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on P(ωω) induced by F turns out to look like the Wadge hierarchy (which is the special case where F is the set of continuous functions).

A Course on Borel Sets

Remarks on Small Sets of Reals

We show that the Dual Borel Conjecture implies that d > א1 and find some topological characterizations of perfectly meager and universally meager 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...

Limitwise monotonic sets of reals

1. f (x, s) f (x, s + 1) for all x and s; 2. sups f (x, s) < ∞ for every x ; 3. F(x) = sups f (x, s). A set A ⊆ N is limitwise monotonic if A equals to the range of some limitwise monotonic function. If we replace here the computable functions f by X -computable functions for some Turing oracle X then we get the notions of X -limitwise monotonic functions and sets, respectively. Note that a sim...