Selection for Borel Relations

Abstract
We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [DSR 96,99,01,01X] involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main results for the reals.
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Alain Louveau (1989). Two Results on Borel Orders. Journal of Symbolic Logic 54 (3):865-874.
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