Abstract
AbstractDefine to be the smallest cardinality of a function f: X→Y with I, 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 < 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 Borei 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.
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