Borel hierarchies in infinite products of Polish spaces

Abstract

Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let ${\Sigma }_{\xi }$ $({\bar{\Sigma }}_{\xi })$ be the class of Borel sets of additive class ξfor the product of copies of the discrete topology on X (the Polish topology on X), and let $\mathcal{B}={\cup }_{\xi <{\omega }_1}{\bar{\Sigma }}_{\xi }$. We prove in the Lévy--Solovay model that \barΣ_ξ=Σ_ξ\cap {\cal B} for $1\le \xi <{\omega }_1$.

7 pages