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Banach-Tarski paradox using pieces with the property of Baire. (English) Zbl 0768.04002

The authors prove the following version of the Banach-Tarski paradox which does not depend on the axiom of choice: If \(A\) and \(B\) are nonempty bounded open subsets of \(\mathbb{R}^ n\), \(n\geq 3\), then there are finitely many pairwise disjoint open subsets \(A_ i\) of \(A\) whose union is dense in \(A\) and isometries \(\rho_ i\) of \(\mathbb{R}^ n\) such that \(B_ i=\rho_ i A_ i\) are pairwise disjoint open subsets of \(B\) whose union is dense. More generally, if \(X\) is a separable metric space and \(G\) is a countable group of homeomorphisms which acts freely on \(X\) such that some \(\{\rho_ i, \gamma_ i: i=1,2\text{ or } 3\}\subseteq G\) generate a free subgroup of rank 6, then there are disjoint open sets \(\{R_ i, G_ i: i=1,2,3\}\) such that \(\bigcup\{\rho_ i R_ i: 1\leq i\leq 3\}\) and \(\bigcup\{\gamma_ i G_ i: 1\leq i\leq 3\}\) are dense open subsets of \(X\). As an application, the authors solve a problem due to E. Marczewski (Problem 1 in S. Wagon: The Banach-Tarski paradox (1985; Zbl 0569.43001)): There is a paradoxical decomposition of the 2-sphere using pieces with the property of Baire. Moreover, the authors announce that the minimal number of 6 such pieces suffice.
Reviewer: N.Brunner (Wien)

MSC:

03E15 Descriptive set theory

Citations:

Zbl 0569.43001
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