Reductions on equivalence relations generated by universal sets

Let X , Y be Polish spaces, Γ ⊆ ℘ ( Y ) , A ⊆ X × Y . We say A is universal for Γ provided that each x ‐section of A is in Γ and each element of Γ occurs as an x ‐section of A . An equivalence relation generated by a set A ⊆ X × Y is denoted by E A , where x E A x ′ ⟺ A x = A x ′. The following results are shown: (1) If A is aΣ n 1 set universal for all nonempty closed subsets of Y , then E A is a σ ( Σ n 1 ) equivalence relation andE A ≤ σ ( Σ n 1 )id ( 2 ω ) . (2) If A is aΣ 1 1 set universal for all countable subsets of Y , then E A is a σ ( Σ 1 1 ) equivalence relation, and (i)E A ≤ σ ( Σ 1 1 )= +and= + ≤ Δ 2 1E A ; (ii) if V = L , thenE A ≤ Δ 2 1id ( 2 ω ) ; (iii) if everyΣ 2 1 set is Lebesgue measurable or has the Baire property, thenE A ≰ Δ 2 1id ( 2 ω ) . (iv) for n ≥ 2 , if everyΔ n 1 set has the Baire property, and E is anyΣ 3 0 equivalence relation, thenE A ≰ Δ n 1 E .