{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:2QP3CJ3VAWGQEPF755SNO7YACI","short_pith_number":"pith:2QP3CJ3V","canonical_record":{"source":{"id":"1507.04087","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-15T05:25:26Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"07655711c28ca4991f08338c2c64928b98df29d2cee7cc0b98276cc84bcae35d","abstract_canon_sha256":"a00583f2e3600bb651ec286bf0d8e8d5bf2d82c505bcbc041677245816040130"},"schema_version":"1.0"},"canonical_sha256":"d41fb12775058d023cbfef64d77f00120b00f06e8e658db40eea821df3f40131","source":{"kind":"arxiv","id":"1507.04087","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04087","created_at":"2026-05-18T01:17:00Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04087v2","created_at":"2026-05-18T01:17:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04087","created_at":"2026-05-18T01:17:00Z"},{"alias_kind":"pith_short_12","alias_value":"2QP3CJ3VAWGQ","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2QP3CJ3VAWGQEPF7","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2QP3CJ3V","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:2QP3CJ3VAWGQEPF755SNO7YACI","target":"record","payload":{"canonical_record":{"source":{"id":"1507.04087","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-15T05:25:26Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"07655711c28ca4991f08338c2c64928b98df29d2cee7cc0b98276cc84bcae35d","abstract_canon_sha256":"a00583f2e3600bb651ec286bf0d8e8d5bf2d82c505bcbc041677245816040130"},"schema_version":"1.0"},"canonical_sha256":"d41fb12775058d023cbfef64d77f00120b00f06e8e658db40eea821df3f40131","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:00.620251Z","signature_b64":"SP8iv2bJ2emgRksHWOjs6/2coA/KWbzNnCW4CGQBoEQJwYSjtXHNhodeUxTu/zHBuh8FnA5xPzWnmMyZLAYqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d41fb12775058d023cbfef64d77f00120b00f06e8e658db40eea821df3f40131","last_reissued_at":"2026-05-18T01:17:00.619641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:00.619641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.04087","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W6NPMpr12/8vMlJQ8IlB87d2b6UJUa6qNBkKZlo2h4GK9ZAjEyksggojN0MTscLgt+nVbGAOGkCP8FO3hmfDAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:56:59.152251Z"},"content_sha256":"b2dc3bad1b2f11085f1704cb2831a53b1036707096427d77f1a14b31b2ce0834","schema_version":"1.0","event_id":"sha256:b2dc3bad1b2f11085f1704cb2831a53b1036707096427d77f1a14b31b2ce0834"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:2QP3CJ3VAWGQEPF755SNO7YACI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two Weak Forms of Countability Axioms in Free Topological Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.GR","authors_text":"Chuan Liu, Fucai Lin, Jiling Cao","submitted_at":"2015-07-15T05:25:26Z","abstract_excerpt":"Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. For every $n\\in\\mathbb{N}$, let $F_{n}(X)$ (resp. $A_n(X)$) denote the subspace of $F(X)$ (resp. $A(X)$) that consists of words of reduced length at most $n$ with respect to the free basis $X$. In this paper, we discuss two weak forms of countability axioms in $F(X)$ or $A(X)$, namely the $csf$-countability and $snf$-countability. We provide some characterizations of the $csf$-countability and $snf$-countability of $F(X)$ and $A(X"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04087","kind":"arxiv","version":2},"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":"2026-05-18T01:17:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SBb9Fea1IgmIfeuV7Yxd473AurKwVhw8aAcmc1etIr//RgkimZME+ATb5TLZ9gavKQ4ntliF/N/rZm13/8wWCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:56:59.152941Z"},"content_sha256":"4680093724728aedac6056a66a95fb2d208d8c4dd870c6581a6aa216f16458b0","schema_version":"1.0","event_id":"sha256:4680093724728aedac6056a66a95fb2d208d8c4dd870c6581a6aa216f16458b0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2QP3CJ3VAWGQEPF755SNO7YACI/bundle.json","state_url":"https://pith.science/pith/2QP3CJ3VAWGQEPF755SNO7YACI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2QP3CJ3VAWGQEPF755SNO7YACI/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":"2026-06-06T19:56:59Z","links":{"resolver":"https://pith.science/pith/2QP3CJ3VAWGQEPF755SNO7YACI","bundle":"https://pith.science/pith/2QP3CJ3VAWGQEPF755SNO7YACI/bundle.json","state":"https://pith.science/pith/2QP3CJ3VAWGQEPF755SNO7YACI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2QP3CJ3VAWGQEPF755SNO7YACI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2QP3CJ3VAWGQEPF755SNO7YACI","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":"a00583f2e3600bb651ec286bf0d8e8d5bf2d82c505bcbc041677245816040130","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-15T05:25:26Z","title_canon_sha256":"07655711c28ca4991f08338c2c64928b98df29d2cee7cc0b98276cc84bcae35d"},"schema_version":"1.0","source":{"id":"1507.04087","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04087","created_at":"2026-05-18T01:17:00Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04087v2","created_at":"2026-05-18T01:17:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04087","created_at":"2026-05-18T01:17:00Z"},{"alias_kind":"pith_short_12","alias_value":"2QP3CJ3VAWGQ","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2QP3CJ3VAWGQEPF7","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2QP3CJ3V","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:4680093724728aedac6056a66a95fb2d208d8c4dd870c6581a6aa216f16458b0","target":"graph","created_at":"2026-05-18T01:17:00Z","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":"Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. For every $n\\in\\mathbb{N}$, let $F_{n}(X)$ (resp. $A_n(X)$) denote the subspace of $F(X)$ (resp. $A(X)$) that consists of words of reduced length at most $n$ with respect to the free basis $X$. In this paper, we discuss two weak forms of countability axioms in $F(X)$ or $A(X)$, namely the $csf$-countability and $snf$-countability. We provide some characterizations of the $csf$-countability and $snf$-countability of $F(X)$ and $A(X","authors_text":"Chuan Liu, Fucai Lin, Jiling Cao","cross_cats":["math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-15T05:25:26Z","title":"Two Weak Forms of Countability Axioms in Free Topological Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04087","kind":"arxiv","version":2},"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:b2dc3bad1b2f11085f1704cb2831a53b1036707096427d77f1a14b31b2ce0834","target":"record","created_at":"2026-05-18T01:17:00Z","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":"a00583f2e3600bb651ec286bf0d8e8d5bf2d82c505bcbc041677245816040130","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-15T05:25:26Z","title_canon_sha256":"07655711c28ca4991f08338c2c64928b98df29d2cee7cc0b98276cc84bcae35d"},"schema_version":"1.0","source":{"id":"1507.04087","kind":"arxiv","version":2}},"canonical_sha256":"d41fb12775058d023cbfef64d77f00120b00f06e8e658db40eea821df3f40131","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d41fb12775058d023cbfef64d77f00120b00f06e8e658db40eea821df3f40131","first_computed_at":"2026-05-18T01:17:00.619641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:00.619641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SP8iv2bJ2emgRksHWOjs6/2coA/KWbzNnCW4CGQBoEQJwYSjtXHNhodeUxTu/zHBuh8FnA5xPzWnmMyZLAYqBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:00.620251Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.04087","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2dc3bad1b2f11085f1704cb2831a53b1036707096427d77f1a14b31b2ce0834","sha256:4680093724728aedac6056a66a95fb2d208d8c4dd870c6581a6aa216f16458b0"],"state_sha256":"a372cf50d6211a043c050741e03b3e31cdae0b137791e88a7fb09c1cd2ee44af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qzJDn5bCs9BupqdpFkCfT+yQKbt+1ROELSQZdTBKr1ZBoxwv+d9CxLFW1aCjd627f8i/NkNYEOqctymcNv9uDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T19:56:59.156469Z","bundle_sha256":"8c3ea022db3734835f2457d69654473766d1e3ac1ffa46f003d92352718c59f5"}}