Ideal topologies in higher descriptive set theory (Q2667997)

The article presents investigations of the topology of the higher Cantor space on \(2^\kappa\). Different topologies on \(2^\kappa\) are considered, induced by ideals other than the bounded ideals on \(\kappa\). The main focus is on the topology (called the nonstationary topology or the Edinburgh topology) induced by the nonstationary ideal on \(2^\kappa\). For a \(<\!\!\kappa \)-complete proper ideal \(\mathcal{I}\) on \(\kappa\) the \(\mathcal{I}\)-topology is defined. The class of Borel sets of ideal topologies is also studied. One of the main questions left open is whether there is an \(\mathcal{I}\)-Borel hierarchy which resembles the usual Borel hierarchy on the Cantor space. Possible notions of convergence, accumulation points, subsequences in ideal topologies, the connection between ideal topologies and forcing topologies are investigated. In the last sections of the article the authors investigate the connections between meager and \(\mathcal{I}\)-meager sets, and between the Baire property in the bounded and in the nonstationary topology.