  For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite
subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for
every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto
\{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on
$X^n$.
  We study the completion of the metric space $\mathsf F^1\!X=(\mathsf
FX,d^1_{\mathsf FX})$ 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.

General Topology;Metric Geometry