Forcing, games and families of closed sets

Sabok, Marcin

doi:10.1090/S0002-9947-2012-05404-3

Forcing, games and families of closed sets
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Trans. Amer. Math. Soc. 364 (2012), 4011-4039 Request permission

Abstract:

We study forcing properties of $\sigma$-ideals generated by closed sets. We show that if a $\sigma$-ideal is $\mathbf {\Pi }^1_1$ on $\mathbf {\Sigma }^1_1$ and generated by closed sets, then it is generated by closed sets in all forcing extensions. This implies that the countable-support iteration of forcings associated with such $\sigma$-ideals is proper. We use it to prove an infinite-dimensional version of the Solecki theorem about inscribing positive $\mathbf {G}_\delta$ sets into positive analytic sets.

We also propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. We provide a tree representation of such forcings, which generalizes the classical approach to Sacks and Miller forcing.

Among the examples, we investigate the $\sigma$-ideal $\mathcal {E}$ generated by closed null sets and $\sigma$-ideals connected with not piecewise continuous functions. For the first one we show that the associated forcing extensions are of minimal degree. For the second one we show that the associated forcing notion is equivalent to Miller forcing.

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