{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XO2LAWBEAU35ZDSKTTRZ3VISSI","short_pith_number":"pith:XO2LAWBE","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"},"canonical_sha256":"bbb4b058240537dc8e4a9ce39dd5129223985834de57e89f3e82292be2de4c96","source":{"kind":"arxiv","id":"1610.02522","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XO2LAWBEAU35ZDSKTTRZ3VISSI","target":"record","payload":{"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"},"canonical_sha256":"bbb4b058240537dc8e4a9ce39dd5129223985834de57e89f3e82292be2de4c96","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"},"source_kind":"arxiv","source_id":"1610.02522","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+f2n8C3fdQlzdPgMD0EPlOTwufYl1hF2UMKgKW973T4+a+++dx8AYivy8xCXXXAJiafhJIqon9S8H1nLlXUmBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T17:20:21.441629Z"},"content_sha256":"a4f9fda57cb7e7b651fd0471de81efa838a23c22278fc12eea82d3f209cd67b0","schema_version":"1.0","event_id":"sha256:a4f9fda57cb7e7b651fd0471de81efa838a23c22278fc12eea82d3f209cd67b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XO2LAWBEAU35ZDSKTTRZ3VISSI","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lfsgvmbCoekqRo4Je1QUYcPG7LHkJVpiXOut0iW8o964H3hwK1Oocd9V6CJfr1mW3vVwrDgnEUNeqb7fFA0HAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T17:20:21.442247Z"},"content_sha256":"bc769d7a5fe61219002a7ed34875322602d90811937b4eb73edf90a169edd63d","schema_version":"1.0","event_id":"sha256:bc769d7a5fe61219002a7ed34875322602d90811937b4eb73edf90a169edd63d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/bundle.json","state_url":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T17:20:21Z","links":{"resolver":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI","bundle":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/bundle.json","state":"https://pith.science/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XO2LAWBEAU35ZDSKTTRZ3VISSI/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k6sPz6r1eMa0JPk1o4Py9MhumEzlyAdrEkyrnYdnNm/WYdbVpAHFBFABzQaZ7cOJYvdhOZxzXba/WphaO27CBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T17:20:21.446085Z","bundle_sha256":"f1e0e53d7adc3472cacd680ffdae24ca0ec3961d4ee9c86fcaa0e6d46a4d0894"}}