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arXiv · 2611.02531

No Borel upper bound for the $F_σ$ equivalence relations

Ruijun Wang
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Abstract

We prove that the family of coset relations of $F_σ$ real linear subspaces of Banach space $\ell_1$ 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.

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Ruijun Wang. 2026-11-02. No Borel upper bound for the $F_σ$ equivalence relations. https://arxiv.org/abs/2611.02531

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