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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hausdorff ultrafilters


Authors: Mauro Di Nasso and Marco Forti
Journal: Proc. Amer. Math. Soc. 134 (2006), 1809-1818
MSC (2000): Primary 03E05, 03H05, 54D80
Published electronically: 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 ($ 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.


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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
Published electronically: 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 28 years after publication.