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Register a new userThe completion of the hyperspace of finite subsets, endowed with the $ell^1$-metric
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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 $ninmathbb N$ the map $X^ntomathsf FX$, $(x_1,dots,x_n)mapsto {x_1,dots,x_n}$, 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 $nge 2$ the Euclidean space $mathbb R^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.
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The Vietoris hyperspace $NC^{*}(X)$ of noncut subcontinua of a metric continuum $X$ has been previously studied by several authors. In this paper we prove that if $X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrational numbers.
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