{"paper":{"title":"Hausdorff hyperspaces of $R^m$ and their dense subspaces","license":"","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Katsuro Sakai, Wieslaw Kubis","submitted_at":"2007-12-05T17:23:38Z","abstract_excerpt":"Let $CLB_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CLB_H(R^m)$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space $\\ell_2$.\n  Moreover, we investigate the hyperspace $CL_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of $CL_H(R)$ is homeomorphic to the Hilbert space $\\el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.0786","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"}