Determinateness and the separation property

Extract

A pointclass is a class of subsets of the Baire space (^ωω) closed under inverse images by continuous functions. The dual of a pointclass Γ, denoted is {~A∣ A ∈ Γ}. (Complements are relative to ^ωω.) If Γ is nonselfdual, i.e. , then let . We say a nonselfdual pointclass Γ has the first separation property, and write Sep(Γ), iff (∀A, B ∈ Γ)(A ⋂ B = ∅ ⇒ (∃C ∈ Δ)(A ⊆ C ∧ B ⋂ = ∅)). The set C is said to separate A and B.

Descriptive set theory abounds in nonselfdual pointclasses Γ, and for the more natural examples of such Γ one can always show by assuming enough determinateness that exactly one of Sep(Γ) and Sep() holds. Van Wesep [2] provides a partial explanation of this fact by showing that, assuming the full axiom of determinateness, one of Sep(Γ) and Sep() must fail for all nonselfdual pointclasses Γ. We shall complete the explanation by showing that one of Sep(Γ) and Sep() must hold.

The axiom of determinateness has other interesting consequences in the general theory of pointclasses. See e.g. [3].