{"paper":{"title":"Distance sets of universal and Urysohn metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Norbert Sauer","submitted_at":"2011-07-24T20:12:53Z","abstract_excerpt":"A metric space $\\mathrm{M}=(M;\\de)$ is {\\em homogeneous} if for every isometry $f$ of a finite subspace of $\\mathrm{M}$ to a subspace of $\\mathrm{M}$ there exists an isometry of $\\mathrm{M}$ onto $\\mathrm{M}$ extending $f$. A metric space $\\boldsymbol{U}$ is an {\\em Urysohn} metric space if it is homogeneous and separable and complete and if it isometrically embeds every separable metric space $\\mathrm{M}$ with $\\dist(\\mathrm{M})\\subseteq \\dist(\\boldsymbol{U})$. (With $\\dist(\\mathrm{M})$ being the set of distances between points in $\\mathrm{M}$.)\n  The main results are: (1) A characterization "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.4794","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}