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Some points in βN

Published online by Cambridge University Press:  24 October 2008

Kenneth Kunen
Kenneth Kunen
Affiliation:
University of Wisconsin
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Abstract

Assuming either the continuum hypothesis or Martin's axiom, we show that in the space βN – N, there are: (a) points which are not P-points, but which are also not limit points of any countable set, and (b) a countable set of points dense in itself such that each of the points is not a limit point of any countable discrete set. Our method is to construct such points in the Stone space of a measure algebra, and then embed that Stone space into βN – N. We also, by a similar use of measures, establish the independence of the existence of selective ultrafilters by showing that there are none in the random real model.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 3 , November 1976 , pp. 385 - 398
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Booth, D.Ultrafilters on a countable set. Ann. Math. Logic 2 (1970), 1–24.CrossRefGoogle Scholar
(2)Comfort, W. W. and Negrepontis, S.The Theory of Ultrafilters (Springer-Verlag; Berlin, 1974).CrossRefGoogle Scholar
(3)Efimov, B. A.Extremally disconnected compact spaces and absolutes. Trudy Moskov. Mat. Obši. 23 (1970). Trans. Moscow Math. Soc. 23 (1970), 243–285.Google Scholar
(4)Fichtenholz, G. and Kantorovitch, L.Sur les opérations linéairs dans l'espace des fonctions bornées. Studia Math. 5 (1934), 69–98.CrossRefGoogle Scholar
(5)Halmos, P. R.Measure theory (D. Van Nostrand Co., Inc.; New York, 1950).CrossRefGoogle Scholar
(6)Halmos, P. R.Naïve set theory (Van Nostrand Reinhold Co.; New York, 1960).Google Scholar
(7)Halmos, P. R.Lectures on Boolean algebras (D. Van Nostrand Co., Inc.; Princeton 1963).Google Scholar
(8)Jech, T. J.Lectures in set theory (Springer Verlag; Berlin, 1971).Google Scholar
(9)Kunen, K. Inaccessibility properties of cardinals. Doctoral dissertation (Stanford, 1968).Google Scholar
(10)Kunen, K.Ultrafilters and independent sets. Trans. Amer. Math. Soc. 172 (1972), 299–306.CrossRefGoogle Scholar
(11)Kunen, K. On normality of box products of ordinals (to appear).Google Scholar
(12)Kunen, K. and Tall, F. D. Between Martin's axiom and Suslin's hypothesis (to appear).Google Scholar
(13)Martin, D. A. and Solovay, R. M.Internal Cohen extensions. Ann. Math. Logic 2 (1970), 143–178.CrossRefGoogle Scholar
(14)Michael, E., Olson, R. C. and Siwiec, F. A-spaces and countably bi-quotient maps (to appear).Google Scholar
(15)Rudin, M. E.Partial orders on the types of β N. Trans. Amer. Math. Soc. 155 (1971), 353–362.Google Scholar
(16)Rudin, W.Homogeneity problems in the theory of Cech, compactifications. Duke Math. J. 23 (1956), 409–419.CrossRefGoogle Scholar
(17)Solovay, R. M.Real-valued measurable cardinals, in Axiomatic Set Theory, A.M.S. Symposia Vol. XIII, Part I (1971), pp. 397–428.Google Scholar