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Ultrafilters with property $ ({s})$

Author: Arnold W. Miller
Journal: Proc. Amer. Math. Soc. 137 (2009), 3115-3121
MSC (2000): Primary 03E35, 03E17, 03E50
Published electronically: April 20, 2009
MathSciNet review: 2506470
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Abstract: A set $ X\subseteq 2^\omega$ has property (s) (Marczewski (Szpilrajn)) iff for every perfect set $ P\subseteq 2^\omega$ there exists a perfect set $ Q\subseteq P$ such that $ Q\subseteq X$ or $ Q\cap X=\emptyset$. Suppose $ {\mathcal{U}}$ is a nonprincipal ultrafilter on $ \omega$. It is not difficult to see that if $ {\mathcal{U}}$ 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 $ {\mathcal{U}}$ 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 $ {\mathcal{U}}$ with property (s) such that $ {\mathcal{U}}$ does not generate an ultrafilter in any extension which adds a new subset of $ \omega$.

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

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.