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Combinatorics for the dominating and unsplitting numbers. (English) Zbl 1069.03038
Summary: In this paper we introduce a new property of families of functions on the Baire space, called pseudo-dominating, and apply the properties of these families to the study of cardinal characteristics of the continuum. We show that the minimum cardinality of a pseudo-dominating family is \(\min\{\mathfrak{r,d}\}\). We derive two corollaries from the proof: \(\mathfrak r\geq \min\{\mathfrak{d,u}\}\) and \(\min\{\mathfrak{d,r}\}= \min\{\mathfrak{d,r}_\sigma\}\). We show that if a dominating family is partitioned into fewer than \(\mathfrak s\) pieces, then one of the pieces is pseudo-dominating. We finally show that \(\mathfrak u < \mathfrak g\) implies that every unbounded family of functions is pseudo-dominating, and that the Filter Dichotomy principle is equivalent to every unbounded family of functions being finitely pseudo-dominating.

MSC:
03E17 Cardinal characteristics of the continuum
03E05 Other combinatorial set theory
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