The ideal determined by the unsymmetric game

Ludomir Newelski and Andrzej Roslanowski
 
 
We study the $\sigma$-ideal of subsets of the Baire space $\omega^\omega$ for which the second player has a winning strategy in the unsymmetric game (i.e. in the game in which Player I is allowed to play finite sequences of integers while the second player answers by single integers). We prove that the additivity number of this ideal is $\omega_1$ and that the cofinality number (the minimal size of the basis of the ideal) is $2^\omega$. Related ideals on compact spaces $n^\omega$ (for $2<n<\omega$) are studied and some consistency results are obtained. Some new forcing notions appear naturally here as well as new localization properties of extensions of models of ZFC (like: ``each real $r\in\omega^\omega$ in the extension is an $\omega$-branch in a tree $T\subseteq\omega^{<\omega}$ from the ground model such that each node in the tree has at most n successors'').
Andrzej Roslanowski
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