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Hausdorff ultrafilters

Authors: Mauro Di Nasso and Marco Forti
Journal: Proc. Amer. Math. Soc. 134 (2006), 1809-1818
MSC (2000): Primary 03E05, 03H05, 54D80
Posted: January 4, 2006
MathSciNet review: 2207497
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Abstract: We give the name Hausdorff to those ultrafilters that provide ultrapowers whose natural topology (-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.

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Additional Information

Mauro Di Nasso
Affiliation: Dipartimento di Matematica ``L. Tonelli'', Università di Pisa, Italy
Email: dinasso@dm.unipi.it

Marco Forti
Affiliation: Dipartimento di Matematica Applicata ``U. Dini'', Università di Pisa, Italy
Email: forti@dma.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08433-4
PII: S 0002-9939(06)08433-4
Received by editor(s): November 24, 2003
Received by editor(s) in revised form: May 12, 2004
Posted: January 4, 2006
Additional Notes: This work was partially supported by the MIUR PRIN Grant ``Metodi logici nello studio di strutture geometriche, topologiche e insiemistiche'', Italy.
Communicated by: Alan Dow
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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