Skip to content
DraconiticResearch
Search Sources API
190,869 studies31 sourcesUpdated 1 h ago
PreprintCC BY 4.0arXiv

No Borel upper bound for the Fσ equivalence relations

  • Ruijun Wang
Open at sourcePDFDOICite

Abstract

We prove that the family of coset relations of Fσ real linear subspaces of Banach space ℓ1 has no common upper bound among Borel equivalence relations in the sense of Borel reducibility. These relations are indexed by a well-founded tree. Consequently, on a Polish space, a universal Σ0α equivalence relation exists exactly when α=1, and a universal Π0α equivalence relation exists exactly when α=1 or $2$. This strengthens the earlier nonexistence theorem for a universal Fσ equivalence relation. This also answers a question of Kanovei and Lyubetsky about universal equivalence relations in Borel classes.

Related

  1. Realizations of Countable Borel Equivalence Relations
    Joshua Frisch, Alexander S. Kechris, Forte Shinko +1
    Memoirs of the American Mathematical SocietyPeer-reviewedLicence not statedFull text8 citedGeometry & topologyOpenAlex
  2. Finite-index extensions of essentially free countable Borel equivalence relations
    Jason Zesheng Chen
    arXivPreprintCC BY 4.0Full textLogic & foundationsarXiv
  3. Borel-Tukey morphisms between Ramsey homogeneity relations
    Tatsuya Goto
    arXivPreprintarXiv licenceFull textLogic & foundationsarXiv
  4. Finite index extensions of treeable equivalence relations are treeable
    Ruiyuan Chen, Anush Tserunyan, Robin Tucker-Drob
    arXivPreprintarXiv licenceFull textAlgebra & number theoryDynamical systemsarXiv
  5. Idealistic equivalence relations and the Lusin derivative
    Bilge Köksal, Mateusz Lichman, Sławomir Solecki
    arXivPreprintCC BY 4.0Full textLogic & foundationsarXiv

Cite

BibTeXRIS
BibTeX
@misc{wang2026borel,
  title = {{No Borel upper bound for the $F_σ$ equivalence relations}},
  author = {Ruijun Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02531},
  eprint = {2611.02531},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02531},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - No Borel upper bound for the F_σ equivalence relations
AU  - Ruijun Wang
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02531
UR  - https://arxiv.org/abs/2611.02531
AB  - We prove that the family of coset relations of F_σ real linear subspaces of Banach space ℓ_1 has no common upper bound among Borel equivalence relations in the sense of Borel reducibility. These relations are indexed by a well-founded tree. Consequently, on a Polish space, a universal Σ^0_α equivalence relation exists exactly when α=1, and a universal Π^0_α equivalence relation exists exactly when α=1 or $2$. This strengthens the earlier nonexistence theorem for a universal F_σ equivalence relation. This also answers a question of Kanovei and Lyubetsky about universal equivalence relations in Borel classes.
ER  -