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Saturating ultrafilters on N

Published online by Cambridge University Press:  12 March 2014

D. H. Fremlin  and
P. J. Nyikos
D. H. Fremlin
Affiliation:
Department of Mathematics, University of Essex, Wivenhoe Park, Colchester C04 3SQ, England
P. J. Nyikos
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
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Abstract

We discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω, c)-saturating ultrafllter on N iff 2λ ≤ c for every λ < c and there is no cover of R by fewer than c nowhere dense sets, (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafllter on N is (ω, κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω, κ)-saturating, and (ii) the set of (ω, κ)-saturating ultrafilters is invariant under homeomorphisms of βN/N. (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω, c)-saturating.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 708 - 718
Copyright
Copyright © Association for Symbolic Logic 1989

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