{"paper":{"title":"Metric spaces admitting only trivial weak contractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Rich\\'ard Balka","submitted_at":"2012-02-07T21:15:25Z","abstract_excerpt":"If $(X,d)$ is a metric space then the map $f\\colon X\\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\\in X$, $x\\neq y$. We determine the simplest non-closed sets $X\\subseteq \\mathbb{R}^n$ in the sense of descriptive set theoretic complexity such that every weak contraction $f\\colon X\\to X$ is constant. In order to do so, we prove that there exists a non-closed $F_{\\sigma}$ set $F\\subseteq \\mathbb{R}$ such that every weak contraction $f\\colon F\\to F$ is constant. Similarly, there exists a non-closed $G_{\\delta}$ set $G\\subseteq \\mathbb{R}$ such that every weak con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1539","kind":"arxiv","version":3},"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"}