Banach-Tarski paradox using pieces with the property of Baire.

Abstract

In 1924 Banach and Tarski, using ideas of Hausdorff, proved that there is a partition of the unit sphere S2 into sets A1,...,Ak,B1,..., Bl and a collection of isometries [sigma1,..., sigmak, rho1,..., rhol] so that [sigma1A1,..., sigmakAk] and [rho1B1,..., rholBl] both are partitions of S2. The sets in these partitions are constructed by using the axiom of choice and cannot all be Lebesgue measurable. In this note we solve a problem of Marczewski from 1930 by showing that there is a partition of S2 into sets A1,..., Ak, B1,..., Bl with a different strong regularity property, the Property of Baire. We also prove a version of the Banach-Tarski paradox that involves only open sets and does not use the axiom of choice.