On Borel sets in ideal topologies

TL;DR

This paper investigates the structure of Borel and analytic sets within ideal topologies on uncountable product spaces, demonstrating the non-collapse of the Borel hierarchy and the equivalence of Borel and analytic sets under certain conditions.

Contribution

It proves the non-collapse of the Borel hierarchy and the analyticity of all Borel sets in ideal topologies, extending understanding of descriptive set theory in these contexts.

Findings

Borel hierarchy does not collapse in ideal topologies.

Every Borel set is analytic in these topologies.

If the ideal contains an unbounded set, all sets are analytic.

Abstract

We study the Borel and analytic subsets of the spaces \({}^{\kappa}\kappa\) and \({}^{\kappa}2\) endowed with ideal topologies, where \(\kappa\) is a regular uncountable cardinal. We establish that the Borel hierarchy does not collapse in any ideal topology and prove that every Borel set in such a topology is analytic. In particular, when the ideal contains an unbounded set, the class of analytic sets coincides with the entire powerset. Furthermore, we show that the Approximation Lemma holds for ideal topologies.