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Determinateness and the separation property

  • John R. Steel (a1)

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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 {~AAΓ}. (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, BΓ)(AB = ∅ ⇒ (∃CΔ)(ACB ⋂ = ∅)). 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].

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References

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[1]Kechris, A. S., Classifying projective-like hierarchies, Bulletin of the Greek Mathematical Society, vol. 18 (1979), pp. 254275.
[2]Van Wesep, R., Separation principles and the axiom of determinateness, this Journal, vol. 43 (1978), pp. 7781.
[3]Van Wesep, R., Wadge degrees and descriptive set theory, Cabal Seminar 76–77, Springer Lecture Notes in Mathematics, vol. 689 (Kechris, A. S. and Moschovakis, Y. N., Editors), Springer-Verlag, Berlin and New York, 1978.

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