On Borel sets in ideal topologies

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.