On the Class of Perfectly Null Sets and Its Transitive Version

Volume 64 / 2016

Michał Korch, Tomasz Weiss Bulletin Polish Acad. Sci. Math. 64 (2016), 1-20 MSC: 03E05, 03E15, 03E35, 03E20. DOI: 10.4064/ba8029-5-2016 Published online: 17 June 2016


We introduce two new classes of special subsets of the real line: the class of perfectly null sets and the class of sets which are perfectly null in the transitive sense. These classes may play the role of duals to the corresponding classes on the category side. We investigate their properties and, in particular, we prove that every strongly null set is perfectly null in the transitive sense, and that it is consistent with ZFC that there exists a universally null set which is not perfectly null in the transitive sense. Finally, we state some open questions concerning the above classes. Although the main problem of whether the classes of perfectly null sets and universally null sets are consistently different remains open, we prove some results related to this question.


  • Michał KorchInstitute of Mathematics
    University of Warsaw
    02-097 Warszawa, Poland
  • Tomasz WeissInstitute of Mathematics
    Cardinal Stefan Wyszyński University in Warsaw
    01-938 Warszawa, Poland

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