Ideal Topologies in Higher Descriptive Set Theory

TL;DR

This paper explores new topologies on higher Cantor spaces based on ideals like the nonstationary ideal, revealing properties of the nonstationary topology and its implications for forcing axioms.

Contribution

It introduces the nonstationary topology on $2^$, extending the understanding of higher topologies and shows $$-Silver forcing satisfies a strong Axiom $A$ under $$-diamond.

Findings

The nonstationary topology differs from the bounded ideal topology.

$$-Silver forcing satisfies a strong form of Axiom $A$ under $$-diamond.

The topology may be of independent interest in 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 $\diamondsuit_\kappa$.