There is no universal Fσ equivalence relation
There is no universal Fσ equivalence relation. This is the main result of arxiv:2610.31052. More detail: An equivalence relation E on a Polish space X is said to be Fσ if it is a countable union of closed subsets of X × X. For equivalence relations E and F on Polish spaces X and Y, we say that E is Borel reducible to F, written E ≤_B F, if there exists a Borel function f : X → Y such that for all x, y ∈ X, we have x E y if and only if f(x) F f(y). An Fσ equivalence relation E is universal if for every Fσ equivalence relation F, we have F ≤_B E. The theorem states that there's no such relation.
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