Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara

Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara

Fundamenta Mathematicae (2015)

Volume: 230, Issue: 1, page 1-13
ISSN: 0016-2736

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Abstract

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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

Δ ⁰_{β + 1}

domains such that γ + α ≤ β if and only if the preimage of each

Σ_{α + 1}^{0}

set under that function is

Σ_{β + 1}^{0}

, and the transformation of a

Σ_{α + 1}^{0}

set into the

Σ_{β + 1}^{0}

preimage is continuous.

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Takayuki Kihara. "Decomposing Borel functions using the Shore-Slaman join theorem." Fundamenta Mathematicae 230.1 (2015): 1-13. <http://eudml.org/doc/282605>.

@article{TakayukiKihara2015,
abstract = {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.},
author = {Takayuki Kihara},
journal = {Fundamenta Mathematicae},
keywords = {Baire function; Borel measurable function; Turing degree},
language = {eng},
number = {1},
pages = {1-13},
title = {Decomposing Borel functions using the Shore-Slaman join theorem},
url = {http://eudml.org/doc/282605},
volume = {230},
year = {2015},
}

TY - JOUR
AU - Takayuki Kihara
TI - Decomposing Borel functions using the Shore-Slaman join theorem
JO - Fundamenta Mathematicae
PY - 2015
VL - 230
IS - 1
SP - 1
EP - 13
AB - 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.
LA - eng
KW - Baire function; Borel measurable function; Turing degree
UR - http://eudml.org/doc/282605
ER -

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