On Borel sets in ideal topologies

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, thereby addressing some open problems of the literature. We provide a systematic analysis of the Borel hierarchy for an arbitrary ideal topology. In particular, we formulate a sufficient condition ensuring that the hierarchy does not collapse, demonstrate that every Borel set in such a topology is analytic, and establish the existence of a set that is not Borel. We show that, when the underlying ideal contains an unbounded subset, the collection of analytic sets coincides with the full power set of the ambient space. Finally, we prove that the Approximation Lemma holds in the setting of ideal topologies.