Combinatorics for the dominating and unsplitting numbers

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{𝖗, 𝖉}. We derive two corollaries from the proof: 𝖗 ≥ min{ 𝖉, 𝖚 } and min{ 𝖉, 𝖗 } = min{ 𝖉, 𝖗_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.