Peter Holy

Ideal and Tree Forcing Topologies

While the usual topology on the kappa-reals is based on the bounded ideal, in the sense that the basic open sets are generated by bounded partial functions from kappa to 2, we consider generalized topologies based on arbitrary ideals on kappa, in particular on the nonstationary ideal on regular and uncountable cardinals kappa. We will illustrate the connections of these topologies with certain tree forcing topologies, and in particular the connection of the nonstationary topology with the topology generated by kappa-Silver forcing. We will show how properties of this forcing notion carry over to properties of the nonstationary topology, and we will also generalize results on kappa-Silver forcing of Friedman, Khomskii and Kulikov. This is joint work with Marlene Koelbing, Philipp Schlicht and Wolfgang Wohofsky.