$U$-Meager Sets when the Cofinality and the Coinitiality of $U$ are Uncountable

Abstract

We prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set Xf = { I, . . , HI, where H is an infinite integer, cannot be expressed as a countable union of countably determined each of which is Umeager for some cut U with min {cf(U), ci(U)} ? co,. Also, every Borel, Z,1 or countably determined set C which is U-meager for every cut U is a countable union of Borel, Z1 or countably determined respectively, which are U-nowhere dense for every cut U. Further, the class of Borel U-meager for min{cf(U),ci(U)} ? wo1 coincides with the least family of containing internal U-meager and closed with respect to the operation of countable union and intersection. The same is true if the phrase Umeager sets is replaced by U-meager for every cut U' Introduction. In [KL] the authors introduced the notion of a U-meager and Unowhere dense set in the set Xf = {1,... , H}, where H is an infinite integer in a nonstandard model of the superstructure over the set of standard integers co. The set U is an initial segment of X closed with respect to addition, and is called a cut. Now, a set M is called U-nowhere dense if, for every interval I of length > U, M has a gap of length > U in I. A set is U-meager if it is a countable union of U-nowhere dense sets. The reader is referred to [KL] for details about the U-meager and U-nowhere dense sets. The purpose of this paper is to try to give some answers to some open questions from [KL], as well as to reveal some properties of U-meager for which the cofinality cf(U) and the coinitiality ci(U) are both uncountable. Here the cofinality of U is defined as the least K such that U has a cofinal subset of cardinality Kc, and the coinitiality of U is the least K such that -$ U has a coinitial subset of cardinality K. The Borel hierarchy of subsets of a hyperfinite set is defined in the usual manner, i.e., internal are called H1 or Z? and, by induction, for every a < co1, a set is HO (Z) if it is a countable intersection (union) of Z1 (HO) for fi < a. A set is Borel if it belongs to the union of the above defined classes. Received May 3, 1990; revised August 29, 1990. ? 1991, Association for Symbolic Logic 0022-4812/91/5603-001 5/$01.90