Ultrafilters with property

Author:
Arnold W. Miller

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3115-3121

MSC (2000):
Primary 03E35, 03E17, 03E50

DOI:
https://doi.org/10.1090/S0002-9939-09-09919-5

Published electronically:
April 20, 2009

MathSciNet review:
2506470

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Abstract | References | Similar Articles | Additional Information

Abstract: A set has property (s) (Marczewski (Szpilrajn)) iff for every perfect set there exists a perfect set such that or . Suppose is a nonprincipal ultrafilter on . It is not difficult to see that if is preserved by Sacks forcing, i.e., if it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then has property (s) in the ground model. It is known that selective ultrafilters or even P-points are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that assuming CH (or more generally MA) there exists an ultrafilter with property (s) such that does not generate an ultrafilter in any extension which adds a new subset of .

**1.**Bartoszyński, Tomek; Judah, Haim;**Set theory. On the structure of the real line.**A K Peters, Ltd., Wellesley, MA, 1995. (Ultrafilters can't be measurable or have property of Baire.) MR**1350295 (96k:03002)****2.**Baumgartner, James E.; Laver, Richard; Iterated perfect-set forcing. Ann. Math. Logic 17 (1979), no. 3, 271-288. MR**556894 (81a:03050)****3.**Blass, Andreas; A partition theorem for perfect sets. Proc. Amer. Math. Soc. 82 (1981), no. 2, 271-277. MR**609665 (83k:03063)****4.**Brendle, Jorg; Between -points and nowhere dense ultrafilters. Israel J. Math. 113 (1999), 205-230. MR**1729447 (2000m:03117)****5.**Kechris, Alexander S.;**Classical descriptive set theory.**Graduate Texts in Mathematics, 156. Springer-Verlag, New York, 1995. MR**1321597 (96e:03057)****6.**Miller, Arnold W.; Rational perfect set forcing.**Axiomatic set theory**(Boulder, Colo., 1983), 143-159, Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984. MR**763899 (86f:03084)****7.**Miller, Arnold W.; Special subsets of the real line.**Handbook of set-theoretic topology**, 201-233, North-Holland, Amsterdam, 1984. MR**776624 (86i:54037)****8.**Mycielski, Jan; Independent sets in topological algebras. Fund. Math. 55 (1964), 139-147. MR**0173645 (30:3855)****9.**Sacks, Gerald E.; Forcing with perfect closed sets.**Axiomatic Set Theory**(Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), pp. 331-355, Amer. Math. Soc., Providence, RI, 1971. MR**0276079 (43:1827)****10.**Solovay, Robert M.; On the cardinality of sets of reals.**Foundations of Mathematics**(Symposium Commemorating Kurt Gödel, Columbus, Ohio, 1966), pp. 58-73, Springer, New York, 1969. MR**0277382 (43:3115)**

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Additional Information

**Arnold W. Miller**

Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388

Email:
miller@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-09919-5

Received by editor(s):
October 27, 2003

Received by editor(s) in revised form:
January 15, 2004

Published electronically:
April 20, 2009

Additional Notes:
Thanks to the Fields Institute, Toronto, for their support during the time these results were proved and to Juris Steprans for helpful conversations, and thanks to Boise State University for support during the time this paper was written

Communicated by:
Alan Dow

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.