{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XO2LAWBEAU35ZDSKTTRZ3VISSI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"649239f8913782915a7dfcd1e03197b70aea1b9b39fd7b9605527a848b64a4ce","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-08T12:57:48Z","title_canon_sha256":"a9526f10480298c665f4894c51affa75f940384cf60ea279b996800901d5bdd1"},"schema_version":"1.0","source":{"id":"1610.02522","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.02522","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1610.02522v6","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02522","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"XO2LAWBEAU35","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XO2LAWBEAU35ZDSK","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XO2LAWBE","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:bc769d7a5fe61219002a7ed34875322602d90811937b4eb73edf90a169edd63d","target":"graph","created_at":"2026-05-18T00:29:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Chuan Liu, Fucai Lin, Shou Lin","cross_cats":["math.FA","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-08T12:57:48Z","title":"The $k_{R}$-property of free Abelian topological groups and products of sequential fans"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02522","kind":"arxiv","version":6},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a4f9fda57cb7e7b651fd0471de81efa838a23c22278fc12eea82d3f209cd67b0","target":"record","created_at":"2026-05-18T00:29:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"649239f8913782915a7dfcd1e03197b70aea1b9b39fd7b9605527a848b64a4ce","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-08T12:57:48Z","title_canon_sha256":"a9526f10480298c665f4894c51affa75f940384cf60ea279b996800901d5bdd1"},"schema_version":"1.0","source":{"id":"1610.02522","kind":"arxiv","version":6}},"canonical_sha256":"bbb4b058240537dc8e4a9ce39dd5129223985834de57e89f3e82292be2de4c96","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bbb4b058240537dc8e4a9ce39dd5129223985834de57e89f3e82292be2de4c96","first_computed_at":"2026-05-18T00:29:39.506536Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:39.506536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6Yj2lmNgOFoATlQ/1qXXCOnnd/9xTB42SZnG8shMt8ln8I/E9oKztg2O5a1gdGbZ3Jm+riZuFmDnRgSrPadaBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:39.506983Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.02522","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a4f9fda57cb7e7b651fd0471de81efa838a23c22278fc12eea82d3f209cd67b0","sha256:bc769d7a5fe61219002a7ed34875322602d90811937b4eb73edf90a169edd63d"],"state_sha256":"854e8b00bdf50d3e0d2afb8d75dbfd2c338aaae8f3c13c3b0724aa8c8d64c406"}