Combinatorial properties of the ideal ${\mathfrak P}_2$

Jacek Cichon, Andrzej Roslanowski, Juris Steprans and Bogdan Weglorz
We investigate the ideal ${\mathfrak P}_2$ of subsets A of the Cantor space $2^\omega$ such that for every infinite set $T\subseteq \omega$ the restriction $A\restriction T=\{x\in 2^T: (\exists y\in A)(y\restrictionT=x)\}$ is a proper subset of 2T. We present several new results concerning cardinal invariants of the ideal. For example, we show that the covering number ${\rm cov}({\mathfrak P}_2)$ (= the minimal size of a family of sets from the ideal covering the whole space) of this ideal is not greater than the successor of the cofinality number (the minimal size of a basis) of the null ideal. A new property of forcing notions (New Set - New Function property) appears naturally in this context. The property seems to be a relative of weak distributivity (if looked at from the point of view of Boolean algebras).
Andrzej Roslanowski
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