Some uniformization results
H. Sarbadhikari (1977)
Fundamenta Mathematicae
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H. Sarbadhikari (1977)
Fundamenta Mathematicae
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Rae Michael Shortt (1987)
Czechoslovak Mathematical Journal
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Greg Hjorth, Alexander S. Kechris (2001)
Fundamenta Mathematicae
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Let E₀ be the Vitali equivalence relation and E₃ the product of countably many copies of E₀. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E₃, either E is reducible to E₀ or else E₃ is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible...
R. Purves (1966)
Fundamenta Mathematicae
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K. Musiał (1973)
Colloquium Mathematicae
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Alessandro Andretta, Donald A. Martin (2003)
Fundamenta Mathematicae
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Two sets of reals are Borel equivalent if one is the Borel pre-image of the other, and a Borel-Wadge degree is a collection of pairwise Borel equivalent subsets of ℝ. In this note we investigate the structure of Borel-Wadge degrees under the assumption of the Axiom of Determinacy.
Vassilios Gregoriades (2012)
Fundamenta Mathematicae
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We present the effective version of the theorem about turning Borel sets in Polish spaces into clopen sets while preserving the Borel structure of the underlying space. We show that under some conditions the emerging parameters can be chosen in a hyperarithmetical way and using this we obtain some uniformity results.
Alexey Ostrovsky (2011)
Fundamenta Mathematicae
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Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f: X → Y be a continuous function onto Y ⊂ C with compact preimages of points. If the image f(U) of every clopen set U is the intersection of an open and a closed set, then Y is a Borel set of the same class α. This result generalizes similar results for open and closed functions.
Alexey Ostrovsky (2005)
Acta Universitatis Carolinae. Mathematica et Physica
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Verónica Becher, Pablo Ariel Heiber, Theodore A. Slaman (2014)
Fundamenta Mathematicae
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We show that the set of absolutely normal numbers is Π⁰₃-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π⁰₃-complete in the effective Borel hierarchy.
Benjamin D. Miller (2007)
Fundamenta Mathematicae
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Answering a question of Kłopotowski, Nadkarni, Sarbadhikari, and Srivastava, we characterize the Borel sets S ⊆ X × Y with the property that every Borel function f: S → ℂ is of the form f(x,y) = u(x) + v(y), where u: X → ℂ and v: Y → ℂ are Borel.
Czesław Ryll-Nardzewski (1964)
Fundamenta Mathematicae
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Su Gao, Steve Jackson, Vincent Kieftenbeld (2008)
Fundamenta Mathematicae
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We consider the Borel structures on ordinals generated by their order topologies and provide a complete classification of all ordinals up to Borel isomorphism in ZFC. We also consider the same classification problem in the context of AD and give a partial answer for ordinals ≤ω₂.