F U N D a M E N T a Mathematicae 164 (2000) Borel and Baire Reducibility

We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures. Introduction. The Borel reducibility theory of Polish equivalence relations, at least in its present form, was initiated independently in [FS89] and [HKL90]. There is now an extensive literature on this topic, including fundamental work on the Glimm–Effros dichotomy in [HKL90], on countable Borel equivalence relations in [DJK94], and on Polish group actions in [BK96]. A Polish space is a topological space that is separable and completely metrizable. The Borel subsets of a Polish space form the least σ-algebra containing the open subsets. A Borel function from one Polish space to another is a function such that the inverse image of every open set is Borel. Two Polish spaces are Borel isomorphic if and only if there is a one-one onto Borel function from the first onto the second. This is an equivalence relation. Any two uncountable Polish spaces are Borel isomorphic. See [Ke94]. We also consider Baire measurable subsets of a Polish space. A nowhere dense set in a Polish space is a set whose closure contains no nonempty open set. A meager subset of a Polish space is a set which is the countable union of nowhere dense sets. A Baire (measurable) subset of a Polish space is a set whose symmetric difference with some open set is meager. A comeager subset of a Polish space is a subset whose complement is meager. We have the fundamental Baire category theorem: A Polish space is not meager. A function from one Polish space to another is said to be Baire if and only if the inverse image of every open set is Baire. 2000 Mathematics Subject Classification: 03E15, 54H05.