Degrees of complexity of subsets of the baire space

Let ω be the set of natural numbers, let ω be {α | α : ω → ω}, and let A and B range over the set of subsets of ω. For any A, B: let A be simpler than B (or A ≤ B) iff A = f−1(B) for some f : ω → ω continuous in the Baire topology (i.e., recursive in some δ ∈ ω); let the degree of complexity of A (or dg(A)) be {B | A ≤ B & B ≤ A}; let −A be {α ∈ ω | α 6∈ A}; let dg(A) be dg(A) ∪ dg(−A); and let dg(A) dg(B) if A ≤ B or −A ≤ B. We work in ZF with the axiom of dependent choices; let AD and ACn be the axioms of determinateness (see Fund. Math. 53(1964), 205–224) and constructibility.