Projective subsets of separable metric spaces

Annals of Pure and Applied Logic 50 (1):53-69 (1990)
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Abstract

In this paper we will consider two possible definitions of projective subsets of a separable metric space X. A set A subset of or equal to X is Σ11 iff there exists a complete separable metric space Y and Borel set B subset of or equal to X × Y such that A = {x ε X : there existsy ε Y ε B}. Except for the fact that X may not be completely metrizable, this is the classical definition of analytic set and hence has many equivalent definitions, for example, A is Σ11 iff A is relatively analytic in X, i.e., A is the restriction to X of an analytic set in the completion of X. Another definition of projective we denote by ΣX1 or abstract projective subset of X. A set of A subset of or equal to X is ΣX1 iff there exists an n ε ω and a Borel set B subset of or equal to X × Xn such that A = {x ε X:there existsy ε Xn ε B}. These sets ca n be far more pathological. While the family of sets Σ11 is closed under countable intersections and countable unions, there is a consistent example of a separable metric space X where ΣX1 is not closed under countable intersections or countable unions. This takes place in the Cohen real model. Assuming CH, there exists a separable metric space X such that every Σ11 set is Borel in X but there exists a Σ11 set which is not Borel in X2. The space X2 has Borel subsets of arbitrarily large rank while X has bounded Borel rank. This space is a Luzin set and the technique used here is Steel forcing with tagged trees. We give examples of spaces X illustrating the relationship between Σ11 and ΣX1 and give some consistent examples partially answering an abstract projective hierarchy problem of Ulam

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References found in this work

Analytic Determinacy and 0#. [REVIEW]Leo Harrington - 1978 - Journal of Symbolic Logic 43 (4):685 - 693.
Steel Forcing and Barwise Compactness.Sy D. Friedman - 1982 - Annals of Mathematical Logic 22 (1):31-46.
Forcing with Tagged Trees.John R. Steel - 1978 - Annals of Mathematical Logic 15 (1):55.
Steel Forcing and Barwise Compactness.S. D. Friedman - 1982 - Annals of Mathematical Logic 22 (1):31.
On the Length of Borel Hierarchies.Arnorld W. Miller - 1979 - Annals of Mathematical Logic 16 (3):233.

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