{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:74NDR5JESE63HQXEFMSN72OEK7","short_pith_number":"pith:74NDR5JE","canonical_record":{"source":{"id":"2208.07826","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2022-08-16T16:12:20Z","cross_cats_sorted":[],"title_canon_sha256":"57d7d82994be8594747208671468cd57a886d8e62977f2d3fe13881f11b75ff2","abstract_canon_sha256":"6735348f35e5254197d521e7048c3f74c9ca94edfa2e7aff2770c5657cac1976"},"schema_version":"1.0"},"canonical_sha256":"ff1a38f524913db3c2e42b24dfe9c457ce75ebea8dd89eee9ffed424e9fcf305","source":{"kind":"arxiv","id":"2208.07826","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2208.07826","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"arxiv_version","alias_value":"2208.07826v1","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.07826","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"pith_short_12","alias_value":"74NDR5JESE63","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"pith_short_16","alias_value":"74NDR5JESE63HQXE","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"pith_short_8","alias_value":"74NDR5JE","created_at":"2026-07-05T04:49:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:74NDR5JESE63HQXEFMSN72OEK7","target":"record","payload":{"canonical_record":{"source":{"id":"2208.07826","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2022-08-16T16:12:20Z","cross_cats_sorted":[],"title_canon_sha256":"57d7d82994be8594747208671468cd57a886d8e62977f2d3fe13881f11b75ff2","abstract_canon_sha256":"6735348f35e5254197d521e7048c3f74c9ca94edfa2e7aff2770c5657cac1976"},"schema_version":"1.0"},"canonical_sha256":"ff1a38f524913db3c2e42b24dfe9c457ce75ebea8dd89eee9ffed424e9fcf305","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:49:09.505762Z","signature_b64":"pQhmVE9RqS/6p2Mt9Vy1RDjwhP8iUMZa0qTg3g64VHXfYYmvsF5N0cHPB1+0w7CjTYjj5atKdcXCgoZ+3OnUDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff1a38f524913db3c2e42b24dfe9c457ce75ebea8dd89eee9ffed424e9fcf305","last_reissued_at":"2026-07-05T04:49:09.505297Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:49:09.505297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2208.07826","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":"2026-07-05T04:49:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fH/VFtVCp+snvdE4sgIm9zHObUW76mGVYIS24YqBtEfy2xdXy66QQ1I3qqexXIPNhUF8MHwQ3TIejiGSxF+6Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T19:30:07.321739Z"},"content_sha256":"236839f57f54648a9717d85371a52524a66817cae9cac2dd7796719a88a87241","schema_version":"1.0","event_id":"sha256:236839f57f54648a9717d85371a52524a66817cae9cac2dd7796719a88a87241"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:74NDR5JESE63HQXEFMSN72OEK7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sets completely separated by functions in Bishop Set Theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Iosif Petrakis","submitted_at":"2022-08-16T16:12:20Z","abstract_excerpt":"Within Bishop Set Theory, a reconstruction of Bishop's theory of sets, we study the so-called completely separated sets, that is sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We introduce the notion of a global family of completely separated sets over an index-completely separated set, and we describe its Sigma- and Pi-set. The free completely separated set on a given set is also presented. Purely set-theoretic versions of the classical Stone-\\v{C}ech theorem and the Tychonoff embedding theorem for completely regular spaces are given, r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.07826","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.07826/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T04:49:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oaRqVwFl1nrXKRE7EeWtTCLQQYdUiQS57Cw02eYslMJjdluZQhC7RlGQZSk54NkDKSb8pd/tFm0hO+f2Og0cDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T19:30:07.322239Z"},"content_sha256":"9d6ed61481d7b04d168ad0435edc0fae1ac5af773d0081a0f34fc230d92dfa19","schema_version":"1.0","event_id":"sha256:9d6ed61481d7b04d168ad0435edc0fae1ac5af773d0081a0f34fc230d92dfa19"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/74NDR5JESE63HQXEFMSN72OEK7/bundle.json","state_url":"https://pith.science/pith/74NDR5JESE63HQXEFMSN72OEK7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/74NDR5JESE63HQXEFMSN72OEK7/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":"2026-08-04T19:30:07Z","links":{"resolver":"https://pith.science/pith/74NDR5JESE63HQXEFMSN72OEK7","bundle":"https://pith.science/pith/74NDR5JESE63HQXEFMSN72OEK7/bundle.json","state":"https://pith.science/pith/74NDR5JESE63HQXEFMSN72OEK7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/74NDR5JESE63HQXEFMSN72OEK7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:74NDR5JESE63HQXEFMSN72OEK7","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":"6735348f35e5254197d521e7048c3f74c9ca94edfa2e7aff2770c5657cac1976","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2022-08-16T16:12:20Z","title_canon_sha256":"57d7d82994be8594747208671468cd57a886d8e62977f2d3fe13881f11b75ff2"},"schema_version":"1.0","source":{"id":"2208.07826","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2208.07826","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"arxiv_version","alias_value":"2208.07826v1","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.07826","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"pith_short_12","alias_value":"74NDR5JESE63","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"pith_short_16","alias_value":"74NDR5JESE63HQXE","created_at":"2026-07-05T04:49:09Z"},{"alias_kind":"pith_short_8","alias_value":"74NDR5JE","created_at":"2026-07-05T04:49:09Z"}],"graph_snapshots":[{"event_id":"sha256:9d6ed61481d7b04d168ad0435edc0fae1ac5af773d0081a0f34fc230d92dfa19","target":"graph","created_at":"2026-07-05T04:49:09Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2208.07826/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Within Bishop Set Theory, a reconstruction of Bishop's theory of sets, we study the so-called completely separated sets, that is sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We introduce the notion of a global family of completely separated sets over an index-completely separated set, and we describe its Sigma- and Pi-set. The free completely separated set on a given set is also presented. Purely set-theoretic versions of the classical Stone-\\v{C}ech theorem and the Tychonoff embedding theorem for completely regular spaces are given, r","authors_text":"Iosif Petrakis","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2022-08-16T16:12:20Z","title":"Sets completely separated by functions in Bishop Set Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.07826","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":"sha256:236839f57f54648a9717d85371a52524a66817cae9cac2dd7796719a88a87241","target":"record","created_at":"2026-07-05T04:49:09Z","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":"6735348f35e5254197d521e7048c3f74c9ca94edfa2e7aff2770c5657cac1976","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2022-08-16T16:12:20Z","title_canon_sha256":"57d7d82994be8594747208671468cd57a886d8e62977f2d3fe13881f11b75ff2"},"schema_version":"1.0","source":{"id":"2208.07826","kind":"arxiv","version":1}},"canonical_sha256":"ff1a38f524913db3c2e42b24dfe9c457ce75ebea8dd89eee9ffed424e9fcf305","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff1a38f524913db3c2e42b24dfe9c457ce75ebea8dd89eee9ffed424e9fcf305","first_computed_at":"2026-07-05T04:49:09.505297Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:49:09.505297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pQhmVE9RqS/6p2Mt9Vy1RDjwhP8iUMZa0qTg3g64VHXfYYmvsF5N0cHPB1+0w7CjTYjj5atKdcXCgoZ+3OnUDw==","signature_status":"signed_v1","signed_at":"2026-07-05T04:49:09.505762Z","signed_message":"canonical_sha256_bytes"},"source_id":"2208.07826","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:236839f57f54648a9717d85371a52524a66817cae9cac2dd7796719a88a87241","sha256:9d6ed61481d7b04d168ad0435edc0fae1ac5af773d0081a0f34fc230d92dfa19"],"state_sha256":"073f8d43d4100e453c6c3dca32cedb602e3e3e8229438e97c17654ebbeb51935"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H9kZArTgwc4Y0ajXYVH8moUqJtutoL6BU8bit9ossysgZ/DAzhhdl8j+c4g7qius+P5hONdHR/opYbDYzvPFBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T19:30:07.328207Z","bundle_sha256":"a2cb4e9fa9089212997361dfbe741a7ccb979f563881be0c51f6d5fd8b8b7a5b"}}