2 00 3 Ultrafilters with property ( s )

A set X ⊆ 2 has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. 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 ω. It is a well known classical result due to Sierpinski (see [1]) that a nonprincipal ultrafilter U on ω when considered as a subset of P (ω) = 2 cannot have the property of Baire or be Lebesgue measurable. Here we identify 2 and P (ω) by identifying a subset of ω with its characteristic function. Another very weak regularity property is property (s) of Marczewski (see Miller [7]). A set of reals X ⊆ 2 has property (s) iff for every perfect set P there exists a subperfect set Q ⊆ P such that either Q ⊆ X or Q ∩X = ∅. Here by perfect we mean homeomorphic to 2. It is natural to ask: Question. (Steprans) Can a nonprincipal ultrafilter U have property (s)? If U is an ultrafilter in a model of set theory V and W ⊇ V is another model of set theory then we say U generates an ultrafilter in W if for every z ∈ P (ω) ∩W there exists x ∈ U with x ⊆ z or x ∩ z = ∅. This means that the filter generated by U (i.e. closing under supersets) is an ultrafilter in W . We begin with the following result: 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. Mathematics Subject Classification 2000: 03E35; 03E17; 03E50