The completion of the hyperspace of finite subsets, endowed with the ${\ell }^1$-metric

Abstract

For a metric space $X$, let $\mathsf{F}X$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d_{\mathsf{F}X}^1$ such that for every $n\in \mathbb{N}$ the map $X^n\to \mathsf{F}X$, $(x_1,\dots ,x_n)\mapsto \{x_1,\dots ,x_n\}$, is non-expanding with respect to the ${\ell }^1$-metric on $X^n$. We study the completion of the metric space ${\mathsf{F}}^{1}X=(\mathsf{F}X,d_{\mathsf{F}X}^1)$ and prove that it coincides with the space ${\mathsf{Z}}^{1}X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space ${\mathbb{R}}^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.

12 pages