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Transactions of the American Mathematical Society

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A separation theorem for $\Sigma ^{1}_{1}$ sets


Author: Alain Louveau
Journal: Trans. Amer. Math. Soc. 260 (1980), 363-378
MSC: Primary 04A15; Secondary 03E15, 26A21, 28A05, 54H05
DOI: https://doi.org/10.1090/S0002-9947-1980-0574785-X
MathSciNet review: 574785
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Abstract: In this paper, we show that the notion of Borel class is, roughly speaking, an effective notion. We prove that if a set A is both $\prod _\xi ^0$ and $\Delta _1^1$, it possesses a $\Pi _\xi ^0$-code which is also $\Delta _1^1$. As a by-product of the induction used to prove this result, we also obtain a separation result for $\Sigma _1^1$ sets: If two $\Sigma _1^1$ sets can be separated by a $\Pi _\xi ^0$ set, they can also be separated by a set which is both $\Delta _1^1$ and $\Pi _\xi ^0$. Applications of these results include a study of the effective theory of Borel classes, containing separation and reduction principles, and an effective analog of the Lebesgue-Hausdorff theorem on analytically representable functions. We also give applications to the study of Borel sets and functions with sections of fixed Borel class in product spaces, including a result on the conservation of the Borel class under integration.


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Article copyright: © Copyright 1980 American Mathematical Society