A Normal Form for Borel Sets
extended abstract, not submitted
For each Borel set A of reals, or more generally of infinite strings from a countable alphabet, we obtain a ``normal form'' ofA, by finding a Borel setΩ of maximum simplicity, such that A and Ω continuously reduce to each other.Ω only depend on the equivalence class of A modulo inter-reducibility, moreover, the map: A→Ω is defined in a simple way (in ZF+DC). In case of Borel sets of finite rank, we prove the above result essentially by defining simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base ω1 (under the map which sends every Borel setA of finite rank to its Wadge degree). Extension to transfinite ranks is provided by iterating both ordinal exponentiation and its Borel counterpart. And finally, we show that all these results can be extended to all Borel subsets of Xω when the alphabet X is of any cardinal.
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