23-reducibilityequivrelations

23-reducibilityequivrelations - 1 BOREL AND BAIRE...

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1 BOREL AND BAIRE REDUCIBILITY by Harvey M. Friedman Department of Mathematics Ohio State University Columbus, Ohio 43210 friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ May 25, 1999 INTRODUCTION The Borel reducibility theory of Polish equivalence relations, at least in its present form, was initiated in [FS89]. 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]. Current principal contributors include H. Becker, R. Dougherty, L. Harrington, G. Hjorth, S. Jackson, A.S. Kechris, and A. Louveau, R. Sami, and S. Solecki. 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.
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2 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. A Polish equivalence relation is a pair (X,E), where X is a Polish space and E is an equivalence relation on X. A Borel (analytic) equivalence relation is a Polish equivalence relation (X,E), where E is as a Borel measurable (analytic) subset of X 2 . Let (X,E 1 ) and (Y,E 2 ) be Polish equivalence relations. We say that h:X Y is a reduction from (X,E 1 ) into (Y,E 2 ) if and only if for all x,y X, E 1 (x,y) E 2 (h(x),h(y)). We say that (X,E1) is Borel reducible to (Y,E 2 ) if and only if there is a Borel reduction from (X,E 1 ) to (Y,E 2 ). We also say that (X,E 1 ) is Baire reducible to (Y,E 2 ) if and only if there is a Baire reduction from (X,E 1 ) to (Y,E 2 ). Let STR = (STR, ) be the analytic equivalence relation of all structures of countable similarity type (in the sense of model theory) whose domain is ω , under isomorphism. Also let BR = (P( ω x ω ), ), be the analytic equivalence relation of binary relations on ω under isomorphism. It is well known that BR is not a Borel equivalence relation and BR is Borel isomorphic to STR. It is generally simpler to work with BR. The condition that a Polish equivalence relation E is Borel
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23-reducibilityequivrelations - 1 BOREL AND BAIRE...

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