Decomposing Borel functions using the Shore-Slaman join theorem

• Autorzy: Takayuki Kihara
• Języki publikacji: EN

Abstrakty

EN

Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{σ}$ set under that function is again $F_{σ}$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals α and β with α ≤ β < α·2, every function between Polish spaces having small transfinite inductive dimension is decomposable into countably many Baire class γ functions with $Δ⁰_{β+1}$ domains such that γ + α ≤ β if and only if the preimage of each $Σ^{0}_{α+1}$ set under that function is $Σ^{0}_{β+1}$, and the transformation of a $Σ^{0}_{α+1}$ set into the $Σ^{0}_{β+1}$ preimage is continuous.

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 03E15: Descriptive set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2015
• Tom: 230
• Numer: 1
• Strony: 1-13

Daty

wydano

2015

Twórcy

autor

Takayuki Kihara

• School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292 Japan

Identyfikatory

DOI

10.4064/fm230-1-1