{"paper":{"title":"Topological properties of function spaces over ordinals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GN","authors_text":"Jan Grebik, Jerzy Kakol, Lyubomyr Zdomskyy, Saak Gabriyelyan","submitted_at":"2016-06-13T16:43:57Z","abstract_excerpt":"A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of $C_p(\\kappa)$ and $C_k(\\kappa)$ over ordinals $\\kappa$. We prove that $C_p(\\kappa)$ is always an Ascoli space, while $C_p(\\kappa)$ is a $k_R$-space iff the cofinality of $\\kappa$ is countable. In particular, this provides the first $C_p$-example of an Ascoli space which is not a $k_R$-space, namely $C_p(\\omega_1)$. We show that $C_k(\\kappa)$ is Ascoli iff $cf(\\kappa)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.04025","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"}