Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T05:18:07.370Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Balcar, Bohuslav and Simon, Petr, On minimal π-character of points in extrematty disconnected compact spaces, Topology andits Applications, vol. 41 (1991), pp. 133–145.CrossRefGoogle Scholar
[2]Blass, Andreas, Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Foreman, Matthew, Kanimori, Akihiro, and Magidor, Menachem, editors), Kluwer, to appear.Google Scholar
[3]Blass, Andreas, Groupwise density and related cardinals, Archive for Mathematical Logic, vol. 30 (1990), no. 1, pp. 1–11.CrossRefGoogle Scholar
[4]Blass, Andreas, Nearly adequate sets, Logic and Algebra, Contemporary Mathematics, vol. 302, 2002, pp. 33–48.Google Scholar
[5]Blass, Andreas and Laflamme, Claude, Consistency results about filters and the number of inequivalent growth types, this Journal, vol. 54 (1989), no. 2, pp. 50–56.Google Scholar
[6]Blass, Andreas and Mildenberger, Heike, On the cofinality of ultrapowers, this Journal, vol. 64 (1999), no. 2, pp. 727–736.Google Scholar
[7]Brendle, Jörg, Around splitting and reaping, Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 2, pp. 269–279.Google Scholar
[8]Ketone, Jussin, On the existence of P-points in the Stone-Čech compactification of the integers. Fundamenta Mathematicae, vol. 92 (1976), pp. 92–94.Google Scholar
[9]Laflamme, Claude, Equivalence of families of functions on the natural numbers, Transactions of the American Mathematical Society, vol. 40 (1992), pp. 307–319.CrossRefGoogle Scholar
[10]Mildenberger, Heike, Groupwise dense families, Archive for Mathematical Logic, vol. 330 (2001), pp. 93–112.CrossRefGoogle Scholar
[11]Nyikos, Peter, Special ultrafilters and cofinal subsets of ωω, to appear.Google Scholar
[12]Solomon, R. C., Families of sets and functions, Czechoslovak Mathematical Journal, vol. 27 (1977), pp. 556–559.CrossRefGoogle Scholar
[13]Talagrand, Michel, Compacts de fonctions mesurables et filtres non mesurables, Studia Mathematica, vol. 67 (1980), no. 1, pp. 13–43.CrossRefGoogle Scholar