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Ordering MAD families à la Katětov. (English) Zbl 1055.03027

Summary: An ordering \((\leq_K)\) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size \({\mathfrak c}\) and decreasing chains of length \({\mathfrak c}^+\) below every element. Assuming \({\mathfrak t}={\mathfrak c}\) a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every \(\omega^\omega\)-bounding forcing \(\mathbb P\) of size \(\mathfrak c\) there is a Cohen-destructible, \(\mathbb P\)-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.

MSC:

03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
54B20 Hyperspaces in general topology
03E50 Continuum hypothesis and Martin’s axiom
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References:

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