Pseudo -dominating families of functions.

Abstract

The topics presented in this dissertation are a contribution to the study of cardinal characteristics of the continuum. We introduce a property of families of functions on the natural numbers. Families of functions with this property, called pseudo-dominating, arise in relation to a number of other concepts in the study of cardinal characteristics. We exhibit some instances of pseudo-dominating families and apply the properties of these families to obtain some new results about some common characteristics. The main results concern the following cardinal characteristics: u , the minimum cardinality of a base for a non-principal ultrafilter; d , the dominating number; the unsplitting number, r ; and its variant rs , the s -unsplitting number. We prove that if a dominating family is partitioned into fewer than s pieces, then one of the pieces is 2-pseudo-dominating. We also show that each unsplittable family of infinite subsets <blkbd>N</blkbd> gives rise to a 2-pseudo-dominating family of functions of the same cardinality. This implies that r&ge;minu,d </fen> and r&ge;minrs ,d</fen> .