On relatively analytic and Borel subsets

Define z to be the smallest cardinality of a function f : X → Y withX, Y ⊆ 2 such that there is no Borel function g ⊇ f . In this paper we prove that it is relatively consistent with ZFC to have b < z where b is, as usual, smallest cardinality of an unbounded family in ω. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists X ⊆ 2 such that the Borel order ofX is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin. The following is an equivalent definition of z: z = min{|X| : X ⊆ 2, ∃Y ⊆ X Y is not Borel in X} For one direction we can use for each Y ⊆ X its characteristic function f : X → 2. For the other direction use that a function is Borel iff the inverse image of each basic open set is Borel. The following answers a question raised by Zapletal [5] see appendix A. Theorem 1 It is relatively consistent with ZFC that b < z. Define p ∈ P(A) for A ⊆ 2 iff p is a finite set of consistent sentences of the form: 1. “x ∈ ∩m<ωUnm” where x ∈ A, n ∈ ω, or 2. “x / ∈ Unm” where x ∈ 2 , n,m ∈ ω, or 3. “[s] ⊆ Unm” where s ∈ 2 , n,m ∈ ω. Thanks to University of Florida, Gainesville and to Boise State University, Idaho for their hospitality during the time this paper was written and to J.Zapletal and T.Bartoszynski for some helpful discussions. Mathematics Subject Classification 2000: 03E35; 03E17; 03E15