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Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements
Gabriel Debs, Institut de Mathématiques de Jussieu, Paris, France, and Jean Saint Raymond, Institut de Mathématique de Jessieu, Paris, France
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Memoirs of the American Mathematical Society
2007; 118 pp; softcover
Volume: 187
ISBN-10: 0-8218-3971-3
ISBN-13: 978-0-8218-3971-3
List Price: US$68 Individual Members: US$40.80
Institutional Members: US\$54.40
Order Code: MEMO/187/876

One of the aims of this work is to investigate some natural properties of Borel sets which are undecidable in $$ZFC$$. The authors' starting point is the following elementary, though non-trivial result: Consider $$X \subset 2^\omega\times2^\omega$$, set $$Y=\pi(X)$$, where $$\pi$$ denotes the canonical projection of $$2^\omega\times2^\omega$$ onto the first factor, and suppose that

$$(\star)$$: "Any compact subset of $$Y$$ is the projection of some compact subset of $$X$$".

If moreover $$X$$ is $$\mathbf{\Pi}^0_2$$ then

$$(\star\star)$$: "The restriction of $$\pi$$ to some relatively closed subset of $$X$$ is perfect onto $$Y$$"

it follows that in the present case $$Y$$ is also $$\mathbf{\Pi}^0_2$$. Notice that the reverse implication $$(\star\star)\Rightarrow(\star)$$ holds trivially for any $$X$$ and $$Y$$.

But the implication $$(\star)\Rightarrow (\star\star)$$ for an arbitrary Borel set $$X \subset 2^\omega\times2^\omega$$ is equivalent to the statement "$$\forall \alpha\in \omega^\omega, \,\aleph_1$$ is inaccessible in $$L(\alpha)$$". More precisely The authors prove that the validity of $$(\star)\Rightarrow(\star\star)$$ for all $$X \in \varSigma^0_{1+\xi+1}$$, is equivalent to "$$\aleph_\xi^L<\aleph_1$$". However we shall show independently, that when $$X$$ is Borel one can, in $$ZFC$$, derive from $$(\star)$$ the weaker conclusion that $$Y$$ is also Borel and of the same Baire class as $$X$$. This last result solves an old problem about compact covering mappings.

In fact these results are closely related to the following general boundedness principle Lift$$(X, Y)$$: "If any compact subset of $$Y$$ admits a continuous lifting in $$X$$, then $$Y$$ admits a continuous lifting in $$X$$", where by a lifting of $$Z\subset \pi(X)$$ in $$X$$ we mean a mapping on $$Z$$ whose graph is contained in $$X$$. The main result of this work will give the exact set theoretical strength of this principle depending on the descriptive complexity of $$X$$ and $$Y$$. The authors also prove a similar result for a variation of Lift$$(X, Y)$$ in which "continuous liftings" are replaced by "Borel liftings", and which answers a question of H. Friedman.

Among other applications the authors obtain a complete solution to a problem which goes back to Lusin concerning the existence of $$\mathbf{\Pi}^1_1$$ sets with all constituents in some given class $$\mathbf{\Gamma}$$ of Borel sets, improving earlier results by J. Stern and R. Sami.

The proof of the main result will rely on a nontrivial representation of Borel sets (in $$ZFC$$) of a new type, involving a large amount of "abstract algebra". This representation was initially developed for the purposes of this proof, but has several other applications.