Combinatorics for the dominating and unsplitting numbers

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{∂, τσ}. 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 thatu<gimplies 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.