On Small Subsets in Euclidean Spaces

Jan Mycielski ¹   ; Grzegorz Tomkowicz ²

¹ Department of Mathematics University of Colorado Boulder, CO 80309-0395, U.S.A.
² Centrum Edukacji $G^2$ Moniuszki 9 41-902 Bytom, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 109-118

Résumé

We study a property of smallness of sets which is stronger than the possibility of packing the set into arbitrarily small balls (i.e., being Tarski null) but weaker than paradoxical decomposability (i.e., being a disjoint union of two sets equivalent by finite decomposition to the whole). We show, using the Axiom of Choice for uncountable families, that there are Tarski null sets which are not small sets. Using only the Principle of Dependent Choices, we show that bounded subsets of $\mathbb {R}^n$ that are included in countable unions of proper analytic subsets of $\mathbb {R}^n$ are small, and several related results.

Détail

DOI : 10.4064/ba8085-10-2016

Keywords: study property smallness sets which stronger possibility packing set arbitrarily small balls being tarski null weaker paradoxical decomposability being disjoint union sets equivalent finite decomposition whole using axiom choice uncountable families there tarski null sets which small sets using only principle dependent choices bounded subsets mathbb included countable unions proper analytic subsets mathbb small several related results