PreprintCC BY 4.0arXiv
There is no universal Fσ equivalence relation
Abstract
Rosendal proved that any Kσ equivalence relation is Borel reducible to ℓ∞. We prove that for every Fσ equivalence relation F on a Polish space, there is an Fσ real linear subspace H of ℓ2 such that the coset relation xmathrelEHy if and only if x−y∈ H has no Baire-measurable reduction to F. In particular, there is no Fσ equivalence relation universal under Borel reducibility for all Fσ equivalence relations. This answers a question of Rosendal.
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BibTeX
@misc{wang2026there,
title = {{There is no universal $F_σ$ equivalence relation}},
author = {Ruijun Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31052},
eprint = {2610.31052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31052},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - There is no universal F_σ equivalence relation
AU - Ruijun Wang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31052
UR - https://arxiv.org/abs/2610.31052
AB - Rosendal proved that any K_σ equivalence relation is Borel reducible to ℓ_∞. We prove that for every F_σ equivalence relation F on a Polish space, there is an F_σ real linear subspace H of ℓ_2 such that the coset relation xmathrelE_Hy if and only if x−y∈ H has no Baire-measurable reduction to F. In particular, there is no F_σ equivalence relation universal under Borel reducibility for all F_σ equivalence relations. This answers a question of Rosendal.
ER -