### Existence of Borel lifting

K. Musiał (1973)

Colloquium 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.

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.

R. Purves (1966)

Fundamenta Mathematicae

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Benjamin Miller, Christian Rosendal (2010)

Journal of the European Mathematical Society

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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.

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...

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.

Petr Holický (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimeasurable maps f and show that some control on the relative classes of preimages and images of Borel sets under f enables one to get a bound on the absolute class of the range of f. This seems to be of some interest even within separable spaces. ...

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.

B. V. Rao (1970)

Colloquium Mathematicae

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B. V. Rao (1971)

Colloquium Mathematicae

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R. M. Shortt (1987)

Colloquium Mathematicae

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Andrzej Komisarski, Henryk Michalewski, Paweł Milewski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.

Alexey Ostrovsky (2005)

Acta Universitatis Carolinae. Mathematica et Physica

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Czesław Ryll-Nardzewski (1964)

Fundamenta Mathematicae

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Stone, A. H.

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H. Sarbadhikari (1977)

Fundamenta Mathematicae

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