Hausdorff ultrafilters

Di Nasso, Mauro; Forti, Marco

doi:10.1090/S0002-9939-06-08433-4

Hausdorff ultrafilters
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by Mauro Di Nasso and Marco Forti PDF

Proc. Amer. Math. Soc. 134 (2006), 1809-1818 Request permission

Abstract:

We give the name Hausdorff to those ultrafilters that provide ultrapowers whose natural topology ($S$-topology) is Hausdorff, e.g. selective ultrafilters are Hausdorff. Here we give necessary and sufficient conditions for product ultrafilters to be Hausdorff. Moreover we show that no regular ultrafilter over the “small” uncountable cardinal $\mathfrak {u}$ can be Hausdorff. ($\mathfrak {u}$ is the least size of an ultrafilter basis on $\omega$.) We focus on countably incomplete ultrafilters, but our main results also hold for $\kappa$-complete ultrafilters.

References

A. Bartoszynski, S. Shelah, There may be no Hausdorff ultrafilters, manuscript (2003, arXiv:math.LO/0311064).
James E. Baumgartner and Alan D. Taylor, Partition theorems and ultrafilters, Trans. Amer. Math. Soc. 241 (1978), 283–309. MR 491193, DOI 10.1090/S0002-9947-1978-0491193-1
Marco Forti, Mauro Di Nasso, and Vieri Benci, Hausdorff nonstandard extensions, Bol. Soc. Parana. Mat. (3) 20 (2002), no. 1-2, 9–20 (2003). MR 2010860
Andreas Blass, The Rudin-Keisler ordering of $P$-points, Trans. Amer. Math. Soc. 179 (1973), 145–166. MR 354350, DOI 10.1090/S0002-9947-1973-0354350-6
A. Blass, Combinatorial cardinal characteristics of the continuum, to appear in Handbook of Set Theory (M. Foreman, M. Magidor, A. Kanamori, eds.).
A. Blass, G. Moche, Finite preimages under the natural map from $\beta (\mathbb {N}\times \mathbb {N})$ to $\beta \mathbb {N}\times \beta \mathbb {N}$, Topology Proceedings (to appear).
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
Maryvonne Daguenet-Teissier, Ultrafiltres à la façon de Ramsey, Trans. Amer. Math. Soc. 250 (1979), 91–120 (French, with English summary). MR 530045, DOI 10.1090/S0002-9947-1979-0530045-6
Mauro Di Nasso and Marco Forti, Topological and nonstandard extensions, Monatsh. Math. 144 (2005), no. 2, 89–112. MR 2123958, DOI 10.1007/s00605-004-0255-2
T. Jech, Set Theory (3rd edition), Springer, Berlin 2002.
A. Kanamori, Ultrafilters over a measurable cardinal, Ann. Math. Logic 10 (1976), no. 3-4, 315–356. MR 491186, DOI 10.1016/0003-4843(76)90012-7
Akihiro Kanamori, Finest partitions for ultrafilters, J. Symbolic Logic 51 (1986), no. 2, 327–332. MR 840409, DOI 10.2307/2274055
Aki Kanamori and Alan D. Taylor, Separating ultrafilters on uncountable cardinals, Israel J. Math. 47 (1984), no. 2-3, 131–138. MR 738164, DOI 10.1007/BF02760512
Jussi Ketonen, Ultrafilters over measurable cardinals, Fund. Math. 77 (1973), no. 3, 257–269. MR 329897, DOI 10.4064/fm-77-3-257-269
Kenneth Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179–227. MR 277346, DOI 10.1016/0003-4843(70)90013-6
Alan H. Mekler, Donald H. Pelletier, and Alan D. Taylor, A note on a lemma of Shelah concerning stationary sets, Proc. Amer. Math. Soc. 83 (1981), no. 4, 764–768. MR 630051, DOI 10.1090/S0002-9939-1981-0630051-1
Siu-Ah Ng and Hermann Render, The Puritz order and its relationship to the Rudin-Keisler order, Reuniting the antipodes—constructive and nonstandard views of the continuum (Venice, 1999) Synthese Lib., vol. 306, Kluwer Acad. Publ., Dordrecht, 2001, pp. 157–166. MR 1895391
Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622