{"paper":{"title":"Cantor-Bendixson type ranks on Polish spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.LO","authors_text":"Vibeke Quorning","submitted_at":"2018-06-08T15:00:33Z","abstract_excerpt":"For any Polish space $X$ it is well-known that the Cantor-Bendixson rank provides a co-analytic rank on $F_{\\aleph_0}(X)$ if and only if $X$ is a $\\sigma$-compact. In the case of $\\omega^\\omega$ one may recover a co-analytic rank on $F_{\\aleph_0}(\\omega^\\omega)$ by considering the Cantor-Bendixson rank of the induced trees instead. In this paper we will generalize this idea to arbitrary Polish spaces and thereby construct a family of co-analytic ranks on $F_{\\aleph_0}(X)$ for any Polish space $X$. We study the behaviour of this family and compare the ranks to the original Cantor-Bendixson rank"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.03206","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"}