On Borel sets in ideal topologies

2025/12/24 by Miguel Moreno, Moreno, Miguel, Beatrice Pitton +1
Mathematics · Decision Sciences · #Advanced Topology and Set Theory #Advanced Banach Space Theory #Fuzzy and Soft Set Theory

paper · doi:10.48550/arxiv.2512.21140

Abstract

We study the Borel and analytic subsets of the spaces \(^κκ\) and \(^κ2\) endowed with ideal topologies, where \(κ\) 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.

Citations

• Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality
• Generalized Baire class functions
• Shelah's Main Gap and the generalized Borel-reducibility
• Ideal Topologies in Higher Descriptive Set Theory
• Fake reflection
• A descriptive Main Gap Theorem
• Souslin quasi-orders and bi-embeddability of uncountable structures
• On the reducibility of isomorphism relations
• Generalized Descriptive Set Theory and Classification Theory
• Strong dichotomy of cardinality
• Stationary sets and infinitary logic

Related

• Generalized Borel Sets
• Borel Canonization of Analytic Sets with Borel Sections
• Ideal Topologies in Higher Descriptive Set Theory
• Borel and analytic sets in locales
• Topologies derived from the old one via ideals