Abstract

Let X and Y be uncountable Polish spaces. We show in ZF that there is a coanalytic subset P of X × Y with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even PCA = Σ 1 2 , graphs. If X = Y = ω ω , P may be taken to be Π 1 1 . Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic P can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy