Separation principles and the axiom of determinateness

Abstract

Let Γ be a class of subsets of Baire space ( ω ω) closed under inverse images by continuous functions. We say such a Γ is continuously closed . Let , the class dual to Γ, consist of the complements relative to ω ω of members of Γ. If Γ is not selfdual, i.e., , then let . A continuously closed nonselfdual class Γ of subsets of ω Γ is said to have the first separation property [2] if The set C is said to separate A and B . The class Γ is said to have the second separation property [3] if We shall assume the axiom of determinateness and show that if Γ is a continuously closed class of subsets of ω ω and then (1) Γ has the first separation property iff does not have the second separation property, and (2) either Γ or has the second separation property. Of course, (1) and (2) taken together imply that Γ and cannot both have the first separation property.