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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1202.1539","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-07T21:15:25Z","cross_cats_sorted":[],"title_canon_sha256":"01b36490dbb09b263a8baa5ed43a44a8d112987c923e26077ad4fbd1f84363ce","abstract_canon_sha256":"5cdcfba56e73ec82f97e21afa5fffd17a13ca45bb8ad36da71bace7ef92c8217"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:26.421562Z","signature_b64":"18BmQ1JXIJImiln/WAAl2JLFnMMpW6o249/OOKbyclrxS3eSciIYSH3qdkQ+8KuBop63gihmViIgVIAKtaiCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afbbddfe993a9c2f067e378a272cbbb02e8d68d5425f830c25a72c82c8f59a44","last_reissued_at":"2026-05-18T02:41:26.421150Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:26.421150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $x,y\\in X$, $x\\neq y$. 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