On Relatively Analytic and Borel Subsets

Define z to be the smallest cardinality of a function f: X → Y with X. Y ⊆ 2ω 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 ωω. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists X ⊆ 2ω such that the Borel order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin