Continuous images of Borel sets

Continuous Images of Sets of Reals

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.

Continuous vs. Borel Reductions

We present some natural examples of countable Borel equivalence relations E, F with E ≤B F such that there does not exist a continuous reduction from E to F .

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...

Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance

We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature [10] (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct ...