{"paper":{"title":"On countable determination of the Kuratowski measure of noncompactness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Lixin Cheng, Xiaoling Chen","submitted_at":"2021-01-27T15:18:37Z","abstract_excerpt":"A long-standing question in the theory of measures of noncompactness is that for the Kuratowski measure of noncompactness $\\alpha$ defined on a metric space $M$, and for every bounded subset $B\\subset M$, is there a countable subset $B_0\\subset B$ such that $\\alpha(B_0)=\\alpha(B)$? In this paper, we give an affirmative answer to the question above. It is done by showing that for each nonempty set $B$ of a Banach space, there is a countable subset $B_0\\subset B$ so that $B$ is strongly finitely representable in $B_0$, and that there is a free ultrafilter $\\mathcal U$ so that $B$ is affinely iso"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.11481","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.11481/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}