Ideal Topologies in Higher Descriptive Set Theory

Abstract

We investigate generalizations of the topology of the higher Cantor space on $2^\kappa$, based on arbitrary ideals rather than the bounded ideal on $\kappa$. Our main focus is on the topology induced by the nonstationary ideal, and we call this topology the nonstationary topology, or also the Edinburgh topology on $2^\kappa$. It may be of independent interest that as a side result, we show $\kappa$-Silver forcing to satisfy a strong form of Axiom $A$ not only if $\kappa$ is inaccessible (which is well-known), but also under the assumption $\diamondsuit_\kappa$.