The completion of the hyperspace of finite subsets, endowed with the -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.
