Forcing, games and families of closed sets

TL;DR

This paper introduces a game-theoretic approach to idealized forcing using fusion games, extends classical forcing methods, and explores properties of definable $\sigma$-ideals generated by closed sets, including an infinite-dimensional Solecki dichotomy.

Contribution

It generalizes classical forcing techniques via fusion games and proves that certain definable $\sigma$-ideals generated by closed sets remain so in all forcing extensions.

Findings

Fusion game approach generalizes classical forcing methods.

Definable $\sigma$-ideals generated by closed sets are preserved across extensions.

Established an infinite-dimensional Solecki dichotomy for analytic sets.

Abstract

We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable ($\mathbf{\Pi}^1_1$ on $\mathbf{\Sigma}^1_1) $\sigma$-ideals we show that if a $\sigma$-ideal is generated by closed sets, then it is generated by closed sets in all forcing extensions. We also prove an infinite-dimensional version of the Solecki dichotomy for analytic sets. Among examples, we investigate the $\sigma$-ideal $\E$ generated by closed null sets and $\sigma$-ideals connected with not piecewise continuous functions.