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If a normal space X is the union of an increasing sequence of open sets U(1), U(2), U(3) ... such that each U(n) contracts to a point in X, must X be contractible?\n  The main results of the paper are:\n  THEOREM 1. If a normal space X is the union of a sequence of open subsets { U(n) } such that the closure of U(n) is contained in U(n+1) and U(n) contracts to a point in U(n+1) for each n > 0, then X is contractible.\n  COROLLARY 2. If a locally compact sigma-compact normal space X is the union of an increasing sequen"},"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":"1606.05379","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-06-16T22:46:10Z","cross_cats_sorted":["math.AT","math.GT"],"title_canon_sha256":"2399c0a973a4e784726b36ceac8181a366b08b331280e877b03a75724abb78a8","abstract_canon_sha256":"1606580813495d6d3fe15dd833cd163b1945b60d23a15abe5b1fbff5419e4820"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:51.986374Z","signature_b64":"USnTvy0TxiXTrAQ+tbCYfLNdR8QhLUS2GPs/QRzf5M2IDZ+RvFExNfJe8cK0oi4VrRwgNFz1UO/oDyDOS0rTCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07803ab0e98f3c1a057e12ad23a67bac2674282c9505a38d133a6e271f220884","last_reissued_at":"2026-05-18T01:02:51.985882Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:51.985882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Is a monotone union of contractible open sets contractible?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.GT"],"primary_cat":"math.GN","authors_text":"Fredric D. Ancel, Robert D. Edwards","submitted_at":"2016-06-16T22:46:10Z","abstract_excerpt":"This paper presents some partial answers to the following question.\n  QUESTION. If a normal space X is the union of an increasing sequence of open sets U(1), U(2), U(3) ... such that each U(n) contracts to a point in X, must X be contractible?\n  The main results of the paper are:\n  THEOREM 1. If a normal space X is the union of a sequence of open subsets { U(n) } such that the closure of U(n) is contained in U(n+1) and U(n) contracts to a point in U(n+1) for each n > 0, then X is contractible.\n  COROLLARY 2. 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