Borel hierarchies in infinite products of Polish spaces

TL;DR

This paper investigates the structure of Borel hierarchies in infinite products of Polish spaces, establishing a precise relationship between different classes of Borel sets within a specific set-theoretic model.

Contribution

It proves that in the Levy–Solovay model, the classes of Borel sets of additive class ξ coincide with intersections of certain Borel classes, clarifying their structure in infinite product spaces.

Findings

Established the equality ar{ extSigma}_\xi = extSigma_\xi igcap extcal{B} for 1 \leq \xi < \omega_1 in the Levy–Solovay model.

Clarified the structure of Borel hierarchies in infinite products of uncountable Polish spaces.

Provided insights into the relationship between different Borel classes in a set-theoretic context.

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 \xi for the product of copies of the discrete topology on X (the Polish topology on X), and let ${\cal B} = \cup_{\xi < \omega_1} \bar{\Sigma}_\xi$. We prove in the L\'{e}vy--Solovay model that \bar{\Sigma}_\xi =\Sigma_{\xi}\cap {\cal B} for $1 \leq \xi < \omega_1$.