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Displaying similar documents to “On the Borel class of the derived set operator. II”

Preservation of the Borel class under open-LC functions

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.

A classification of ordinals up to Borel isomorphism

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.

Recent developments in the theory of Borel reducibility

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...

Borel-Wadge degrees

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.

Coordinatewise decomposition of group-valued Borel functions

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.

Normal numbers and the Borel hierarchy

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.