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Combinatorics for the dominating and unsplitting numbers

Published online by Cambridge University Press:  12 March 2014

Jason Aubrey
Jason Aubrey*
Affiliation:
Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, USA, E-mail: aubreyj@math.uni.edu
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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 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 2 , June 2004 , pp. 482 - 498
Copyright
Copyright © Association for Symbolic Logic 2004

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