On Small Subsets in Euclidean Spaces

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.