{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:XO2LAWBEAU35ZDSKTTRZ3VISSI","short_pith_number":"pith:XO2LAWBE","schema_version":"1.0","canonical_sha256":"bbb4b058240537dc8e4a9ce39dd5129223985834de57e89f3e82292be2de4c96","source":{"kind":"arxiv","id":"1610.02522","version":6},"attestation_state":"computed","paper":{"title":"The $k_{R}$-property of free Abelian topological groups and products of sequential fans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GR"],"primary_cat":"math.GN","authors_text":"Chuan Liu, Fucai Lin, Shou Lin","submitted_at":"2016-10-08T12:57:48Z","abstract_excerpt":"A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessary and sufficient condition for a real-valued function $f$ on $X$ to be continuous is that the restriction of $f$ to each compact subset is continuous. In this paper, we discuss the $k_{R}$-property of products of sequential fans and free Abelian topological groups by applying the $\\kappa$-fan introduced by Banakh. In particular, we prove the following two results:\n  (1) The space $S_{\\omega_{1}}\\times S_{\\omega_{1}}$ is not a $k_{R}$-space.\n  (2) The space $S_{\\omega}\\times S_{\\omega_{1}}$ is a $k_{R}$-space if and only"},"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":"1610.02522","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-08T12:57:48Z","cross_cats_sorted":["math.FA","math.GR"],"title_canon_sha256":"a9526f10480298c665f4894c51affa75f940384cf60ea279b996800901d5bdd1","abstract_canon_sha256":"649239f8913782915a7dfcd1e03197b70aea1b9b39fd7b9605527a848b64a4ce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:39.506983Z","signature_b64":"6Yj2lmNgOFoATlQ/1qXXCOnnd/9xTB42SZnG8shMt8ln8I/E9oKztg2O5a1gdGbZ3Jm+riZuFmDnRgSrPadaBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbb4b058240537dc8e4a9ce39dd5129223985834de57e89f3e82292be2de4c96","last_reissued_at":"2026-05-18T00:29:39.506536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:39.506536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $k_{R}$-property of free Abelian topological groups and products of sequential fans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GR"],"primary_cat":"math.GN","authors_text":"Chuan Liu, Fucai Lin, Shou Lin","submitted_at":"2016-10-08T12:57:48Z","abstract_excerpt":"A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessary and sufficient condition for a real-valued function $f$ on $X$ to be continuous is that the restriction of $f$ to each compact subset is continuous. In this paper, we discuss the $k_{R}$-property of products of sequential fans and free Abelian topological groups by applying the $\\kappa$-fan introduced by Banakh. In particular, we prove the following two results:\n  (1) The space $S_{\\omega_{1}}\\times S_{\\omega_{1}}$ is not a $k_{R}$-space.\n  (2) The space $S_{\\omega}\\times S_{\\omega_{1}}$ is a $k_{R}$-space if and only"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02522","kind":"arxiv","version":6},"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":"1610.02522","created_at":"2026-05-18T00:29:39.506602+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.02522v6","created_at":"2026-05-18T00:29:39.506602+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02522","created_at":"2026-05-18T00:29:39.506602+00:00"},{"alias_kind":"pith_short_12","alias_value":"XO2LAWBEAU35","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"XO2LAWBEAU35ZDSK","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"XO2LAWBE","created_at":"2026-05-18T12:30:51.357362+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/XO2LAWBEAU35ZDSKTTRZ3VISSI","json":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI.json","graph_json":"https://pith.science/api/pith-number/XO2LAWBEAU35ZDSKTTRZ3VISSI/graph.json","events_json":"https://pith.science/api/pith-number/XO2LAWBEAU35ZDSKTTRZ3VISSI/events.json","paper":"https://pith.science/paper/XO2LAWBE"},"agent_actions":{"view_html":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI","download_json":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI.json","view_paper":"https://pith.science/paper/XO2LAWBE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.02522&json=true","fetch_graph":"https://pith.science/api/pith-number/XO2LAWBEAU35ZDSKTTRZ3VISSI/graph.json","fetch_events":"https://pith.science/api/pith-number/XO2LAWBEAU35ZDSKTTRZ3VISSI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/action/storage_attestation","attest_author":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/action/author_attestation","sign_citation":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/action/citation_signature","submit_replication":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/action/replication_record"}},"created_at":"2026-05-18T00:29:39.506602+00:00","updated_at":"2026-05-18T00:29:39.506602+00:00"}