Reductions on equivalence relations generated by universal sets

Let X, Y be Polish spaces, , . 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 is denoted by , where . The following results are shown:

• (1) If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and .
• (2) If A is a set universal for all countable subsets of Y, then is a equivalence relation, and
  • (i) and ;
  • (ii) if , then ;
  • (iii) if every set is Lebesgue measurable or has the Baire property, then .
  • (iv) for , if every set has the Baire property, and E is any equivalence relation, then .