{"paper":{"title":"A note on bornologies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Igor Protasov","submitted_at":"2018-06-12T05:36:25Z","abstract_excerpt":"A bornology on a set $X$ is a family $\\mathcal{B}$ of subsets of $X$ closed under taking subsets, finite unions and such that $\\cup \\mathcal{B}=X$. We prove that, for a bornology $\\mathcal{B}$ on $X$, the following statements are equivalent:\n  (1) there exists a vector topology $\\tau$ on the vector space $\\mathbb{V} (X) $ over $\\mathbb{R}$ such that $\\mathcal{B}$ is the family of all subsets of $X$ bounded in $\\tau$;\n  (2) there exists a uniformity $\\mathcal{U}$ on $X$ such that $\\mathcal{B}$ is the family of all subsets of $X$ totally bounded in $\\mathcal{U}$;\n  (3) for every $Y \\subseteq X$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.04337","kind":"arxiv","version":2},"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"}