## Forcing, games and families of closed sets

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- by Marcin Sabok PDF
- 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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## Additional Information

**Marcin Sabok**- Affiliation: Mathematical Institute, Wrocław University, pl. Grunwaldzki $2\slash 4$, $50$-$384$ Wrocław, Poland
- Email: sabok@math.uni.wroc.pl
- Received by editor(s): October 15, 2009
- Received by editor(s) in revised form: June 18, 2010
- Published electronically: March 26, 2012
- Additional Notes: This research was supported by MNiSW grant N 201 361836
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**364**(2012), 4011-4039 - MSC (2010): Primary 03E15, 28A05, 54H05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05404-3
- MathSciNet review: 2912443