2020/04/04 by Banakh, Iryna, Banakh, Taras, Garbulińska-Wȩgrzyn, Joanna
#05C90 #54B20 #54E35 #54E50 #54F45 #FOS: Mathematics #General Topology (math.GN) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2004.02019
For a metric space X, let \mathsf FX be the space of all nonempty finite subsets of X endowed with the largest metric d1\mathsf FX such that for every n∈\mathbb N the map Xn→\mathsf FX, (x1,…,xn)↦ \x1,…,xn\, is non-expanding with respect to the ℓ1-metric on Xn. We study the completion of the metric space \mathsf F1 X=(\mathsf FX,d1\mathsf FX) and prove that it coincides with the space \mathsf Z1 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≥ 2 the Euclidean space \mathbb Rn contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.