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We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing and we obtain a cardinal invariant such that collapses the continuum to and . Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of . We define two relations and on the set (ωω)Fin of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if -unbounded, well-ordered by , and not -dominating, then there is a nonmeager p-ideal. The existence of such a system follows from Martin's axiom. This is an analogue of the results of , [9, 10] for increasing functions.
We define the ideal with the property that a real omits all Borel sets in the ideal which are coded in a transitive model if and only if it is an amoeba real over this model. We investigate some other properties of this ideal. Strolling through the "amoeba forest" we gain as an application a modification of the proof of the inequality between the additivities of Lebesgue measure and Baire category.
We define a σ-ideal on the set of functions ωω with the property that a real x ∈ ωω is a Hechler real over V if and only if x omits all Borel sets in . In fact we define a topology on ωω related to Hechler forcing such that is the family of first category sets in . We study cardinal invariants of the ideal .
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