Comparing the closed almost disjointness and dominating numbers

TL;DR

This paper explores the relationship between dominating families and maximal almost disjoint families of functions, establishing conditions under which certain maximal families exist with specific properties.

Contribution

It proves that the existence of a dominating family of size ℵ₁ guarantees a maximal almost disjoint family of functions with particular maximality properties.

Findings

Existence of a maximal almost disjoint family from a dominating family of size ℵ₁

Construction of ℵ₁ many compact subsets forming such a family

Maximality with respect to infinite partial functions

Abstract

We prove that if there is a dominating family of size ${\aleph}_{1}$, then there is are ${\aleph}_{1}$ many compact subsets of ${\omega}^{\omega}$ whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.