A Minimal Counterexample To Universal Baireness

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models. ?0. Introduction and statement of results. A set of reals, A c JIR, is universally Baire if it is well-behaved from the view point of category in the following strong sense: For any topological space X and any continuous function f: X IJR, the pointwise preimage of A under f, f -1 [A], has the property of Baire in the space X, i.e., its symmetric difference with some open set in X is meager. These sets were first studied in [FeMaWo] who also proved a characterization in terms of projections of trees which I shall adopt as the official definition for this paper. [A tree on co x (5, where (5 is an ordinal is a set of finite sequences of elements from co x (5 closed under initial segments. The reals are identified as usual with the space of infinite sequences of non-negative integers.] DEFINITION 0. 1. A set A c JR is universally Baire iffor any A, there are trees TI, T2 on co x ( for some (5 such that 1. A = p[TI] ={x JR :f E ( Vn < co(x In,f In) T1}, and 2. Whenever iP is a notion offorcing of size < A, V"P 1p[TI] JR \ p[T2]. Thus the universally Baire sets may be viewed as generalizations of the analytic and co-analyticsets, and indeed all the classical regularity properties (e.g., Lebesgue measurability, property of Baire, perfect set property and Ramsey property, cf. [Ka]) may be established from the above Suslin representation (cf. [FeMaWo]). It is therefore of interest which sets of reals are universally Baire. There can be no answer in general since this question becomes independent of the standard axioms of set theory ZFC just one step past the first level of the projective hierarchy Received March 14, 1997; revised March 26, 1998. MSC2000 Mathematics Subject Classification. Primary 03E15, 03E45, 54H05, 28A05; Secondary 03E55, 03E60.