Ultrafilters with property (s)
by
Arnold W. Miller
University of Wisconsin, Madison

A set X Í 2^w has property (s) (Marczewski (Spzilrajn)) iff for every perfect set P Í 2^w there exists a perfect set Q Í P such that Q Í X or QÇX=Æ.

It is a classical result that for any nonprinciple ultrafilter U on w , if we consider U as a subset of P(w) which we identify with 2^w, then U cannot be Lebesgue measurable or have the property of Baire. Juris Steprans raised the question of whether an ultrafilter can have property (s).

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) 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 w.

Paper reference: arXiv:math.LO/0310438

Date received: February 24, 2003