{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PXQSCVZPIDQO4SH6DFOBDKRGIP","short_pith_number":"pith:PXQSCVZP","canonical_record":{"source":{"id":"1808.10243","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-08-30T11:57:33Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"cce99d8f43bb98eabc3093e9539da45554ba3a948bc9c9ba785ac0634c1c6de7","abstract_canon_sha256":"310f2edcdce3592f553a94b7f6aaf1766ccdc6e1a9b3988657c602201482d55b"},"schema_version":"1.0"},"canonical_sha256":"7de121572f40e0ee48fe195c11aa2643e1a7c82b66b9ec1628565e9eb9d40c89","source":{"kind":"arxiv","id":"1808.10243","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.10243","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"arxiv_version","alias_value":"1808.10243v1","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.10243","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"pith_short_12","alias_value":"PXQSCVZPIDQO","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PXQSCVZPIDQO4SH6","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PXQSCVZP","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PXQSCVZPIDQO4SH6DFOBDKRGIP","target":"record","payload":{"canonical_record":{"source":{"id":"1808.10243","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-08-30T11:57:33Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"cce99d8f43bb98eabc3093e9539da45554ba3a948bc9c9ba785ac0634c1c6de7","abstract_canon_sha256":"310f2edcdce3592f553a94b7f6aaf1766ccdc6e1a9b3988657c602201482d55b"},"schema_version":"1.0"},"canonical_sha256":"7de121572f40e0ee48fe195c11aa2643e1a7c82b66b9ec1628565e9eb9d40c89","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:49.410358Z","signature_b64":"QwTh1e7G2YpqpGQDdsnAOt5NWe/ukY85tKgbAzGZUv/klF1XCi9L9/tDBKjeqn5UgeF7bKfXRPKE7YZw+5KcDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7de121572f40e0ee48fe195c11aa2643e1a7c82b66b9ec1628565e9eb9d40c89","last_reissued_at":"2026-05-18T00:06:49.409597Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:49.409597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1808.10243","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FJb7eA10CYCB0P1m4VHCNw6nYq+mjm+W6/xDLyjXp15GtoGDsjFHts4JDjYFVprIzZqLH8IgtAVYjsv+y97nAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:06:29.747820Z"},"content_sha256":"2658c5a47182276ab433d8d886fbe13b98bc0132b7433f38ae5e2e5394a14a1f","schema_version":"1.0","event_id":"sha256:2658c5a47182276ab433d8d886fbe13b98bc0132b7433f38ae5e2e5394a14a1f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PXQSCVZPIDQO4SH6DFOBDKRGIP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic topology of Polish spaces. II: Axiomatic homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Sergey A. Melikhov","submitted_at":"2018-08-30T11:57:33Z","abstract_excerpt":"Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common generalization of Milnor's two additivity axioms suffice to uniquely characterize (co)homology of closed pairs of Polish spaces (=separable complete metrizable spaces). The proof provides a combinatorial description of the (co)homology of a Polish space in terms of a cellular (co)chain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10243","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rioEXqkf7EZz7j8FTkEMMkpntL4fBZFPKxUAdw5chbMHdTpMKbsiytT+1rQQe8zIJx0NAsyVBfrz9xJgHdieCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:06:29.748160Z"},"content_sha256":"7abd6ef05b3706902873aeb4a532fb69afbaf2dbde2145d5e6db95b7a067dd9b","schema_version":"1.0","event_id":"sha256:7abd6ef05b3706902873aeb4a532fb69afbaf2dbde2145d5e6db95b7a067dd9b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PXQSCVZPIDQO4SH6DFOBDKRGIP/bundle.json","state_url":"https://pith.science/pith/PXQSCVZPIDQO4SH6DFOBDKRGIP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PXQSCVZPIDQO4SH6DFOBDKRGIP/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":"2026-06-25T12:06:29Z","links":{"resolver":"https://pith.science/pith/PXQSCVZPIDQO4SH6DFOBDKRGIP","bundle":"https://pith.science/pith/PXQSCVZPIDQO4SH6DFOBDKRGIP/bundle.json","state":"https://pith.science/pith/PXQSCVZPIDQO4SH6DFOBDKRGIP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PXQSCVZPIDQO4SH6DFOBDKRGIP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PXQSCVZPIDQO4SH6DFOBDKRGIP","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":"310f2edcdce3592f553a94b7f6aaf1766ccdc6e1a9b3988657c602201482d55b","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-08-30T11:57:33Z","title_canon_sha256":"cce99d8f43bb98eabc3093e9539da45554ba3a948bc9c9ba785ac0634c1c6de7"},"schema_version":"1.0","source":{"id":"1808.10243","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.10243","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"arxiv_version","alias_value":"1808.10243v1","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.10243","created_at":"2026-05-18T00:06:49Z"},{"alias_kind":"pith_short_12","alias_value":"PXQSCVZPIDQO","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PXQSCVZPIDQO4SH6","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PXQSCVZP","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:7abd6ef05b3706902873aeb4a532fb69afbaf2dbde2145d5e6db95b7a067dd9b","target":"graph","created_at":"2026-05-18T00:06:49Z","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":"Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common generalization of Milnor's two additivity axioms suffice to uniquely characterize (co)homology of closed pairs of Polish spaces (=separable complete metrizable spaces). The proof provides a combinatorial description of the (co)homology of a Polish space in terms of a cellular (co)chain","authors_text":"Sergey A. Melikhov","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-08-30T11:57:33Z","title":"Algebraic topology of Polish spaces. II: Axiomatic homology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10243","kind":"arxiv","version":1},"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":"sha256:2658c5a47182276ab433d8d886fbe13b98bc0132b7433f38ae5e2e5394a14a1f","target":"record","created_at":"2026-05-18T00:06:49Z","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":"310f2edcdce3592f553a94b7f6aaf1766ccdc6e1a9b3988657c602201482d55b","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-08-30T11:57:33Z","title_canon_sha256":"cce99d8f43bb98eabc3093e9539da45554ba3a948bc9c9ba785ac0634c1c6de7"},"schema_version":"1.0","source":{"id":"1808.10243","kind":"arxiv","version":1}},"canonical_sha256":"7de121572f40e0ee48fe195c11aa2643e1a7c82b66b9ec1628565e9eb9d40c89","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7de121572f40e0ee48fe195c11aa2643e1a7c82b66b9ec1628565e9eb9d40c89","first_computed_at":"2026-05-18T00:06:49.409597Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:49.409597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QwTh1e7G2YpqpGQDdsnAOt5NWe/ukY85tKgbAzGZUv/klF1XCi9L9/tDBKjeqn5UgeF7bKfXRPKE7YZw+5KcDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:49.410358Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.10243","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2658c5a47182276ab433d8d886fbe13b98bc0132b7433f38ae5e2e5394a14a1f","sha256:7abd6ef05b3706902873aeb4a532fb69afbaf2dbde2145d5e6db95b7a067dd9b"],"state_sha256":"e788f442d8b5ecbae3e6432955021624d9599b3451b8ac8db8f4c6b9d781f344"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z9pgNQ06jp9UCQxMeZjwXPBbwQCPdr+2T5zdKJISXs+/Ck8+Uiu7NhERnhz/a9A2U6UTfd8FJf6r5auKFyrCBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T12:06:29.750091Z","bundle_sha256":"12b490e6d72914e45b316d99ff750e27afbbcf8c1a374119b2767b0a756d5590"}}