Absolute $\mathcal F$-Borel classes

TL;DR

This paper explores the differences between $\\mathcal{F}$-Borel classes and their absolute counterparts, providing examples and showing that for separable metrizable spaces, these classes coincide.

Contribution

It offers a detailed comparison between $\\mathcal{F}$-Borel and absolute $\\mathcal{F}$-Borel classes, including new examples and a key result for separable metrizable spaces.

Findings

Examples distinguishing the hierarchies.

$\\mathcal{F}$-Borel classes are absolute in separable metrizable spaces.

Clarification of the relationship between the two hierarchies.

Abstract

We investigate and compare $\mathcal F$-Borel classes and absolute $\mathcal F$-Borel classes. We provide precise examples distinguishing these two hierarchies. We also show that for separable metrizable spaces, $\mathcal F$-Borel classes are automatically absolute.