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Saturating ultrafilters on N

  • D. H. Fremlin (a1) and P. J. Nyikos (a2)


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.



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[1]Bartoszyński, T., Combinatorial aspects of measure and category, Fundamenta Mathematicae, vol. 127(1987), pp. 225239.
[2]Blass, A., The Rudin-Keisler ordering of P-points, Transactions of the American Mathematical Society, vol. 179 (1973), pp. 145166.
[3]Booth, D., Ultrafilters on a countable set, Annals of Mathematical Logic, vol. 2 (1970/1971), pp. 124.
[4]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.
[5]Comfort, W. W. and Neorepontis, S., The theory of ultrafilters, Springer-Verlag, Berlin, 1974.
[6]Császár, Á. (editor), Topology (proceedings of the fourth colloquium, Budapest, 1978). Vols. I, II, Colloquia Mathematica Societatis János Bolyai, vol. 23, North-Holland, Amsterdam, 1980.
[7]Dordal, P., Towers in [ω 1]ωand ωω, preprint, 1986.
[8]Douwen, E. K. van, The integers and topology, in [16], pp. 111168.
[9]Ellentuck, E. and Rucker, R. V. B., Martin's axiom and saturated models, Proceedings of the American Mathematical Society, vol. 34 (1972), pp. 243249.
[10]Fremlin, D. H., Consequences of Martin's axiom, Cambridge University Press, Cambridge, 1984.
[11]Fremlin, D. H., Cichoń's diagram, Séminaire d'initiation à l'analyse (G. Choquet–M. Rogalski–J. Saint-Raymond), 23ème annee: 1983/1984, Université Pierre et Marie Curie (Paris-VI), Paris, 1984, Exposé 5.
[12]Kucia, A. and Szymański, A., Absolute points in βN/N, Czechoslovak Mathematical Journal, vol. 26(1976), pp. 381387.
[13]Kunen, K., Set theory, North-Holland, Amsterdam, 1980.
[14]Kunen, K., Weak p-points in N*, in [6], pp. 741749.
[15]Kunen, K., Random and Cohen reals, in [16[, pp. 887911.
[16]Kunen, K. and Vaughan, J. E. (editors), Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984.
[17]van Mill, J., An introduction to βω, in [16], pp. 503568.
[18]Miller, A. W., Some properties of measure and category, Transactions of the American Mathematical Society, vol. 266 (1981), pp. 93114.
[19]Miller, A. W., A characterization of the least cardinal for which the Baire category theorem fails, Proceedings of the American Mathematical Society, vol. 86 (1982), pp. 498502.
[20]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.
[21]Weiss, W., Versions of Martin's axiom, in [16], pp. 827886.

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Saturating ultrafilters on N

  • D. H. Fremlin (a1) and P. J. Nyikos (a2)


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