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Forcing, games and families of closed sets


Author: Marcin Sabok
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
Published electronically: March 26, 2012
MathSciNet review: 2912443
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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\slash4, 50-384 Wrocław, Poland
Email: sabok@math.uni.wroc.pl

DOI: https://doi.org/10.1090/S0002-9947-2012-05404-3
Keywords: Proper forcing, $𝜎$-ideals
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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