{"paper":{"title":"The Ascoli property for function spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GN","authors_text":"Jan Greb\\'ik, Jerzy Kakol, Lyubomyr Zdomskyy, Saak Gabriyelyan","submitted_at":"2016-06-03T09:24:11Z","abstract_excerpt":"The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of $k_{\\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli, then it is $\\kappa$-Fr\\'echet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is Ascoli iff it is $\\kappa$-Fr'echet-Urysohn. This leads to the following extension of a result of Morishita: If for a \\v{C}ech-complete space $X$ the space $C_p(X)$ is Ascoli, then $X$ is scattered. If $X$ is scattered and stra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01013","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"}