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In particular, if $Y$ is a Lindel\\\"of $\\Sigma$-space, then $w(X)\\leq nw(Y)^\\omega\\leq nw(X)^\\omega$.\n  Better upper bounds for the weight of topological groups are given. For example, if a topological group $H$ contains a dense subgroup $G$ such that $G$ is a Lindel\\\"of $\\Sigma$-space, then $w(H)=w(G)\\leq \\psi(G)^\\omega$. Further, if a Lindel\\\"of $\\Sigma$-space $X$ generates a dense subgroup of a topological group $H$, then $w(H)\\leq 2^{\\psi(X)}$.\n  Seve"},"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":"1509.02874","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-09-09T18:00:33Z","cross_cats_sorted":[],"title_canon_sha256":"74d37312de461986ac0d57d655eb268c65643ae3cb88c846c9e10e6954cc4d56","abstract_canon_sha256":"cdb06fb269e9504f35609e081553b5d5fdcc8af04187088327010fce94048614"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:33.047881Z","signature_b64":"A0k962x56RsjQ4osx69VmsJ08OTW88GkH6dC8/YeevKGulrYod2k+JtltORuQgs7twcdLa1uGqgrSGI1e0XpDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbde9eade1f4160b4e1c59cfd1f25aacca90c09a9561a999cf38864d085e3674","last_reissued_at":"2026-05-18T01:33:33.047172Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:33.047172Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The weight and Lindel\\\"of property in spaces and topological groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Mikhail G. Tkachenko","submitted_at":"2015-09-09T18:00:33Z","abstract_excerpt":"We show that if $Y$ is a dense subspace of a Tychonoff space $X$, then $w(X)\\leq nw(Y)^{Nag(Y)}$, where $Nag(Y)$ is the Nagami number of $Y$. In particular, if $Y$ is a Lindel\\\"of $\\Sigma$-space, then $w(X)\\leq nw(Y)^\\omega\\leq nw(X)^\\omega$.\n  Better upper bounds for the weight of topological groups are given. For example, if a topological group $H$ contains a dense subgroup $G$ such that $G$ is a Lindel\\\"of $\\Sigma$-space, then $w(H)=w(G)\\leq \\psi(G)^\\omega$. Further, if a Lindel\\\"of $\\Sigma$-space $X$ generates a dense subgroup of a topological group $H$, then $w(H)\\leq 2^{\\psi(X)}$.\n  Seve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02874","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.02874","created_at":"2026-05-18T01:33:33.047303+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.02874v1","created_at":"2026-05-18T01:33:33.047303+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02874","created_at":"2026-05-18T01:33:33.047303+00:00"},{"alias_kind":"pith_short_12","alias_value":"7PPJ5LPB6QLA","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"7PPJ5LPB6QLAWTQ4","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"7PPJ5LPB","created_at":"2026-05-18T12:29:10.953037+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT","json":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT.json","graph_json":"https://pith.science/api/pith-number/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/graph.json","events_json":"https://pith.science/api/pith-number/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/events.json","paper":"https://pith.science/paper/7PPJ5LPB"},"agent_actions":{"view_html":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT","download_json":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT.json","view_paper":"https://pith.science/paper/7PPJ5LPB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.02874&json=true","fetch_graph":"https://pith.science/api/pith-number/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/graph.json","fetch_events":"https://pith.science/api/pith-number/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/action/storage_attestation","attest_author":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/action/author_attestation","sign_citation":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/action/citation_signature","submit_replication":"https://pith.science/pith/7PPJ5LPB6QLAWTQ4LHH5D4S2VT/action/replication_record"}},"created_at":"2026-05-18T01:33:33.047303+00:00","updated_at":"2026-05-18T01:33:33.047303+00:00"}