Borel Liftings of Borel Sets - Some Decidable and Undecidable Statements

One of the motivations for this book was a topological problem concerning particular classes of mappings between separable metrizable spaces. We recall that a surjective continuous mapping f from X onto Y is perfect if the inverse image of a compact set in Y is compact in X. We say that f is a compact covering if any compact subset of Y is the direct image of a compact set in X. The mapping f is said to be inductively perfect if there exists a subset X' in X such that f(X') = Y and f restricted to X' is perfect. Obviously, one has that any perfect mapping is inductively perfect and any inductively perfect mapping is a compact covering. A problem raised by Ostrovsky is whether a compact covering image of a Borel space is also Borel. A more general variant of this question is whether any compact covering mapping between two Borel spaces is also inductively perfect. It has turned out that these problems lead to a question on continuous and Borel liftings.

The main result of the book shows a close relation between the lifting property and certain set-theory inequalities. Methods of proof based upon a tree and a double-tree representation of Borel sets are presented in the first and the second chapter. A couple of applications are shown in chapter 3, namely Hurewicz type results and Borel separation results. The next part is devoted to a solution of Ostrovsky's problem mentioned above and to a question by H. Friedman on Borel liftings. Chapter 5 presents an application of the main theorem to special aspects of Lusin's problem on constituents of coanalytic non-Borel sets. The last part contains the proof of the main theorem. The book contains deep and powerful ideas that seem likely to have much more interesting applications. Thus it is strongly recommended for everyone interested in descriptive set theory and its applications.

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