{"paper":{"title":"The normality and bounded growth of balleans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.MG"],"primary_cat":"math.GN","authors_text":"Igor Protasov, Taras Banakh","submitted_at":"2018-10-18T10:15:20Z","abstract_excerpt":"By a ballean we understand a set $X$ endowed with a family of entourages which is a base of some coarse structure on $X$. Given two unbounded ballean $X,Y$ with normal product $X\\times Y$, we prove that the balleans $X,Y$ have bounded growth and the bornology of $X\\times Y$ has a linearly ordered base. A ballean $(X,\\mathcal E_X)$ is defined to have bounded growth if there exists a function $G$ assigning to each point $x\\in X$ a bounded subset $G[x]\\subset X$ so that for any bounded set $B\\subset X$ the union $\\bigcup_{x\\in B}G[x]$ is bounded and for any entourage $E\\in\\mathcal E_X$ there exis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07979","kind":"arxiv","version":4},"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"}