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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Proceedings of the National Academy of Sciences of the United States of America
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
