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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: [email protected]
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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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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