A syntactic approach to Borel functions: some extensions of Louveau’s theorem

Archive for Mathematical Logic 62 (7):1041-1082 (2023)
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Abstract

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class $$\Gamma $$, then its $$\Gamma $$ -code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -function, then one can find its $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau’s theorem for Borel functions.

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References found in this work

Quasi-Polish spaces.Matthew de Brecht - 2013 - Annals of Pure and Applied Logic 164 (3):356-381.
A Wadge hierarchy for second countable spaces.Yann Pequignot - 2015 - Archive for Mathematical Logic 54 (5):659-683.
A q-wadge hierarchy in quasi-polish spaces.Victor Selivanov - 2022 - Journal of Symbolic Logic 87 (2):732-757.

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