On relatively analytic and Borel subsets

Abstract

Define 𝖟 to be the smallest cardinality of a function f : X→ Y with X,Y ⊆ 2^ω 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^ω. This answers a question raised by Zapletal.

We also show that it is relatively consistent with ZFC that there exists X⊆ 2^ω 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.