On Small Subsets in Euclidean Spaces
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 109-118
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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.
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
Affiliations des auteurs :
Jan Mycielski 1 ; Grzegorz Tomkowicz 2
@article{10_4064_ba8085_10_2016,
author = {Jan Mycielski and Grzegorz Tomkowicz},
title = {On {Small} {Subsets} in {Euclidean} {Spaces}},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
pages = {109--118},
publisher = {mathdoc},
volume = {64},
number = {2-3},
year = {2016},
doi = {10.4064/ba8085-10-2016},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/ba8085-10-2016/}
}
TY - JOUR AU - Jan Mycielski AU - Grzegorz Tomkowicz TI - On Small Subsets in Euclidean Spaces JO - Bulletin of the Polish Academy of Sciences. Mathematics PY - 2016 SP - 109 EP - 118 VL - 64 IS - 2-3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/ba8085-10-2016/ DO - 10.4064/ba8085-10-2016 LA - en ID - 10_4064_ba8085_10_2016 ER -
%0 Journal Article %A Jan Mycielski %A Grzegorz Tomkowicz %T On Small Subsets in Euclidean Spaces %J Bulletin of the Polish Academy of Sciences. Mathematics %D 2016 %P 109-118 %V 64 %N 2-3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4064/ba8085-10-2016/ %R 10.4064/ba8085-10-2016 %G en %F 10_4064_ba8085_10_2016
Jan Mycielski; Grzegorz Tomkowicz. On Small Subsets in Euclidean Spaces. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 109-118. doi: 10.4064/ba8085-10-2016
Cité par Sources :