Reductions on equivalence relations generated by universal sets
Reductions on equivalence relations generated by universal sets
复制标题
通用集生成的等价关系的约简
作者：
Longyun Ding;Ping Yu
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 EA , where xEAx′⟺Ax=Ax′ . The following results are shown: (1)If A is a Σn1 set universal for all nonempty closed subsets of Y, then EA is a σ(Σn1) equivalence relation and EA≤σ(Σn1) id (2ω) . (2)If A is a Σ11 set universal for all countable subsets of Y, then EA is a σ(Σ11) equivalence relation, and (i) EA≤σ(Σ11)=+ and =+≤Δ21EA ; (ii)if V=L , then EA≤Δ21 id (2ω) ; (iii)if every Σ21 set is Lebesgue measurable or has the Baire property, then EA≰Δ21 id (2ω) . (iv)for n≥2 , if every Δn1 set has the Baire property, and E is any Σ30 equivalence relation, then EA≰Δn1E .
