The completion of the hyperspace of finite subsets, endowed with the $\ell ^1$-metric
Volume 166 / 2021
Abstract
For a metric space $X$, let $\mathsf FX$ be the space of all non-empty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in \mathbb N $ the map $X^n\to \mathsf FX$, $(x_1,\ldots ,x_n)\mapsto \{x_1,\ldots ,x_n\}$, is non-expanding with respect to the $\ell ^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of non-empty 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.
