2023/04/10 by Serhii Bardyla, Jonathan Cancino-Manríquez, Bardyla, Serhii +5 · 1 citation
Mathematics · #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2304.04651
A family A ⊆ [ω]ω such that for all finite \Xi\i∈ n⊆ \mathcal A and A ∈ A ∖ \Xi\i∈ n, the set A ∖ \bigcupi ∈ n Xi is infinite, is said to be ideal independent. We prove that an ideal independent family A is maximal if and only if \mathcal A is \mathcal J-completely separable and maximal \mathcal J-almost disjoint for a particular ideal \mathcal J on ω. We show that \mathfraku≤\mathfraksmm, where \mathfraksmm is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of \mathfraksmm and \mathfraki. Given an arbitrary set C of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality λ for each λ∈ C, thus establishing the consistency of C⊆ \hboxspec(\mathfraksmm). Assuming CH, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, ωω-bounding, p-point preserving forcing notion and evaluate \mathfraksmm in several well studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mrówka spaces for ideal independent families.