Splitting families and forcing

Abstract

According to [M.S. Kurilić, Cohen-stable families of subsets of the integers, J. Symbolic Logic 66 (1) (2001) 257-270], adding a Cohen real destroys a splitting family S on ω if and only if S is isomorphic to a splitting family on the set of rationals, Q, whose elements have nowhere dense boundaries. Consequently, | S | < cov (M) implies the Cohen-indestructibility of S. Using the methods developed in [J. Brendle, S. Yatabe, Forcing indestructibility of MAD families, Ann. Pure Appl. Logic 132 (2-3) (2005) 271-312] the stability of splitting families in several forcing extensions is characterized in a similar way (roughly speaking, destructible families have members with 'small generalized boundaries' in the space of the reals). Also, it is proved that a splitting family is preserved by the Sacks (respectively: Miller, Laver) forcing if and only if it is preserved by some forcing which adds a new (respectively: an unbounded, a dominating) real. The corresponding hierarchy of splitting families is investigated. © 2006 Elsevier B.V. All rights reserved.