Combinatorics for the dominating and unsplitting numbers

Source abstract

Abstract. 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 { τ, ∂ }. We derive two corollaries from the proof: τ ≥ min{ ∂, u } and min{ ∂, τ } = min{ ∂, τ σ }. We show that if a dominating family is partitioned into fewer that s pieces, then one of the pieces is pseudo-dominating. We finally show that u < 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.