Effective refining of Borel coverings
| Autor: | Gabriel Debs, Jean Saint Raymond |
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| Rok vydání: | 2009 |
| Předmět: | |
| Zdroj: | Transactions of the American Mathematical Society. 361:2831-2869 |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/s0002-9947-09-04930-7 |
| Popis: | Given a countable family ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} of additive or multiplicative Baire classes ( Γ i = Σ ξ i 0 \mathbf {\Gamma }_i=\mathbf {\Sigma }^0_{\xi _i} or Π ξ i 0 \mathbf {\Pi }^0_{\xi _i} ) we investigate the following complexity problem: Let ( A i ) i ∈ I (A_i)_{i\in I} be a Borel covering of ω ω \omega ^\omega and assume that there exists some covering ( B i ) i ∈ I (B_i)_{i\in I} with B i ⊂ A i B_i\subset A_i and B i ∈ Γ i B_i\in \mathbf {\Gamma }_i for all i i ; can one find such a family ( B i ) i ∈ I (B_i)_{i\in I} in Δ 1 1 ( α ) \varDelta ^1_1(\alpha ) where α ∈ ω ω \alpha \in \omega ^\omega is any reasonable code for the families ( A i ) i ∈ I (A_i)_{i\in I} and ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} ? The main result of the paper will give a full characterization of those families ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} for which the answer is positive. For example we will show that this is the case if I I is finite or if all the Baire classes Γ i \mathbf {\Gamma }_i are additive, but in the general case the answer depends on the distribution of the multiplicative Baire classes inside the family ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} . |
| Databáze: | OpenAIRE |
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