Forcing properties of ideals of closed sets

TL;DR

This paper investigates the forcing properties of ideals generated by closed sets on Polish spaces, exploring their connections, combinatorial aspects, and effects on real degrees in forcing extensions.

Contribution

It introduces a new approach linking forcing properties of sigma-ideals to combinatorial properties of associated ideals on countable sets, especially for those generated by closed sets.

Findings

Identifies conditions under which Borel functions are either injective or constant on I-positive sets.

Establishes connections between forcing properties of I and I* ideals.

Analyzes the degrees of reals added in forcing extensions for sigma-ideals generated by closed sets.

Abstract

With every $\sigma$-ideal $I$ on a Polish space we associate the $\sigma$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing notions of Borel sets modulo the respective $\sigma$-ideals $I$ and $I^*$ and find connections between their forcing properties. To this end, we associate to a $\sigma$-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. For $\sigma$-ideals generated by closed sets we also study the degrees of reals added in the forcing extensions. Among corollaries of our results, we get necessary and sufficient conditions for a $\sigma$-ideal $I$ generated by closed sets, under which every Borel function can be restricted to an $I$-positive Borel set on which it is either 1-1 or constant.