### Abstract

We study classes of Borel subsets of the real line R such as levels of the Borel hierarchy and the class of sets that are reducible to the set Q of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on R. Notably, we explore several structural properties of Borel subsets of R that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order ω1, its reverse ω_{1} ^{?} and the poset P(ω)/fin of inclusion modulo finite error. As a consequence of this proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to Q. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In another main result, we introduce a combinatorial property that is shown to characterize those Fσ sets that are reducible to Q. This is applied to construct a minimal set below Q and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.

Original language | English |
---|---|

Pages (from-to) | 209-241 |

Number of pages | 33 |

Journal | Fundamenta Mathematicae |

Volume | 244 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Keywords

- Borel sets
- Partial orders
- Wadge reducibility

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Fundamenta Mathematicae*,

*244*(3), 209-241. https://doi.org/10.4064/fm644-10-2018

**Borel subsets of the real line and continuous reducibility.** / Ikegami, Daisuke; Schlicht, Philipp; Tanaka, Hisao.

Research output: Contribution to journal › Article

*Fundamenta Mathematicae*, vol. 244, no. 3, pp. 209-241. https://doi.org/10.4064/fm644-10-2018

}

TY - JOUR

T1 - Borel subsets of the real line and continuous reducibility

AU - Ikegami, Daisuke

AU - Schlicht, Philipp

AU - Tanaka, Hisao

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We study classes of Borel subsets of the real line R such as levels of the Borel hierarchy and the class of sets that are reducible to the set Q of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on R. Notably, we explore several structural properties of Borel subsets of R that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order ω1, its reverse ω1 ? and the poset P(ω)/fin of inclusion modulo finite error. As a consequence of this proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to Q. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In another main result, we introduce a combinatorial property that is shown to characterize those Fσ sets that are reducible to Q. This is applied to construct a minimal set below Q and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.

AB - We study classes of Borel subsets of the real line R such as levels of the Borel hierarchy and the class of sets that are reducible to the set Q of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on R. Notably, we explore several structural properties of Borel subsets of R that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order ω1, its reverse ω1 ? and the poset P(ω)/fin of inclusion modulo finite error. As a consequence of this proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to Q. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In another main result, we introduce a combinatorial property that is shown to characterize those Fσ sets that are reducible to Q. This is applied to construct a minimal set below Q and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.

KW - Borel sets

KW - Partial orders

KW - Wadge reducibility

UR - http://www.scopus.com/inward/record.url?scp=85059987878&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059987878&partnerID=8YFLogxK

U2 - 10.4064/fm644-10-2018

DO - 10.4064/fm644-10-2018

M3 - Article

VL - 244

SP - 209

EP - 241

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 3

ER -