Combinatorial properties of the ideal
Jacek Cichon, Andrzej Roslanowski, Juris Steprans and Bogdan
We investigate the ideal
of subsets A of the Cantor space
such that for every infinite set
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
(= 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).
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