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Define z to be the smallest cardinality of a function f : X → Y with X, 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 of X 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:

On relatively analytic and Borel subsets