## 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:

 3e+17 Cardinal characteristics of the continuum 300000 Other combinatorial set theory
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### References:

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