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arXiv · 0910.2318

Forcing, games and families of closed sets

Marcin Sabok
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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 ($\mathbfΠ^1_1$ on $\mathbfΣ^1_1) $σ$-ideals we show that if a $σ$-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 $σ$-ideal $\E$ generated by closed null sets and $σ$-ideals connected with not piecewise continuous functions.

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Marcin Sabok. 2009-10-13. Forcing, games and families of closed sets. https://arxiv.org/abs/0910.2318

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