On the Class of Perfectly Null Sets and Its Transitive Version

Michał Korch¹ ; Tomasz Weiss²

¹ Institute of Mathematics University of Warsaw 02-097 Warszawa, Poland
² Institute of Mathematics Cardinal Stefan Wyszyński University in Warsaw 01-938 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 1, pp. 1-20

Résumé

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.

Détail

DOI : 10.4064/ba8029-5-2016

Keywords: introduce classes special subsets real line class perfectly null sets class sets which perfectly null transitive sense these classes may play role duals corresponding classes category side investigate their properties particular prove every strongly null set perfectly null transitive sense consistent zfc there exists universally null set which perfectly null transitive sense finally state questions concerning above classes although main problem whether classes perfectly null sets universally null sets consistently different remains prove results related question