Title: Ultrafilters with property (s)
Author: Arnold W. Miller
Abstract:
A set X which is a subset of the Cantor set has property (s) (Marczewski
(Spzilrajn)) iff for every perfect set P there exists a perfect set Q
contained in P such that Q is a subset of X or Q is disjoint from X.
Suppose U is a nonprincipal ultrafilter on omega. It is not difficult to
see that if U is preserved by Sacks forcing, i.e., it generates an
ultrafilter in the generic extension after forcing with the partial order
of perfect sets, then 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 for ctble posets) there exists
an ultrafilter U with property (s) such that U does not generate an
ultrafilter in any extension which adds a new subset of omega.
LaTeX2e 10 page
http://www.math.wisc.edu/~miller/res/index.html
miller@math.wisc.edu