2014/08/31 by Taras Banakh, Dušan Repovš
Mathematics · #math.GN #math.MG #msc:51F99 #msc:54E35
paper · pdf · doi:10.4064/cm6697-9-2015
published as Coll. Math. 144:2 (2016) 189-202 · arXiv admin note: text overlap with arXiv:1103.5118, arXiv:0908.3687
arxiv created 2019/09/21 · arxiv updated 2019/09/25
For every metric space X we introduce two cardinal characteristics cov^\flat(X) and cov^\sharp(X) describing the capacity of balls in X. We prove that these cardinal characteristics are invariant under coarse equivalence and prove that two ultrametric spaces X,Y are coarsely equivalent if cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y). This result implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if cov^\flat(X)=cov^\sharp(X). Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and only if cov^\sharp(X)=cov^\sharp(Y) if and only if each of these spaces coarsely embeds into the other space. This means that the coarse structure of an isometrically homogeneous ultrametric space X is completely determined by the value of the cardinal cov^\sharp(X)=cov^\flat(X).