{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:GJBKOO5F3NR47AIE4RTFGLN6LM","short_pith_number":"pith:GJBKOO5F","canonical_record":{"source":{"id":"1303.3810","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T15:37:22Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"923971fb03ffa2bd3cac8f56cf7f7c14510de82f9c13f1c9781ff6c4e514362e","abstract_canon_sha256":"9f8d0cf30ffeb0d90d252cdb21b49693f7b471b0178a835436875f8428a36305"},"schema_version":"1.0"},"canonical_sha256":"3242a73ba5db63cf8104e466532dbe5b239b3cf2afeb86667161f52016f95490","source":{"kind":"arxiv","id":"1303.3810","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3810","created_at":"2026-05-18T00:32:21Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3810v2","created_at":"2026-05-18T00:32:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3810","created_at":"2026-05-18T00:32:21Z"},{"alias_kind":"pith_short_12","alias_value":"GJBKOO5F3NR4","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GJBKOO5F3NR47AIE","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GJBKOO5F","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:GJBKOO5F3NR47AIE4RTFGLN6LM","target":"record","payload":{"canonical_record":{"source":{"id":"1303.3810","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T15:37:22Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"923971fb03ffa2bd3cac8f56cf7f7c14510de82f9c13f1c9781ff6c4e514362e","abstract_canon_sha256":"9f8d0cf30ffeb0d90d252cdb21b49693f7b471b0178a835436875f8428a36305"},"schema_version":"1.0"},"canonical_sha256":"3242a73ba5db63cf8104e466532dbe5b239b3cf2afeb86667161f52016f95490","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:21.975551Z","signature_b64":"iXfTgfNmUidDr5e1/vWNDd6J8nJHHazaUN2/SLJxv4Khw7Mvzu69Hej0LVXi619uyIR9QwT/GsLLy8MSHQDpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3242a73ba5db63cf8104e466532dbe5b239b3cf2afeb86667161f52016f95490","last_reissued_at":"2026-05-18T00:32:21.974877Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:21.974877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.3810","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":"2026-05-18T00:32:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XxQTIiY85ZSSChGhYwFLQOzekE2MG35dah0hj9CXUlE9Z1JjlyFKeG7XeOqjroa4ANlvmqQwzSjv2XTrt3O3BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:28:34.960502Z"},"content_sha256":"0b54a57f50e239c16d0ba50fb234bedf1d611544b0817b9a07e32feaee4bc465","schema_version":"1.0","event_id":"sha256:0b54a57f50e239c16d0ba50fb234bedf1d611544b0817b9a07e32feaee4bc465"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:GJBKOO5F3NR47AIE4RTFGLN6LM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Continuous images of Cantor's ternary set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.DS","authors_text":"Fabian Dreher, Tony Samuel","submitted_at":"2013-03-15T15:37:22Z","abstract_excerpt":"The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set $C$. It is well known that there are compact Hausdorff spaces of cardinality equal to that of $C$ that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of $C$. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3810","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":"2026-05-18T00:32:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PbViaUexg6tjoYTd1nUM6TFSdw8iL7Bk/jyWXm+rwdOu7OnjyvH7uDn3P8HeZCn0qveP6EFrtdaB8i5tEctdCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:28:34.961217Z"},"content_sha256":"46556963d4ce050de21b677231f3a28b3fd75b6df45c2bea9bf6d66fcc5de36a","schema_version":"1.0","event_id":"sha256:46556963d4ce050de21b677231f3a28b3fd75b6df45c2bea9bf6d66fcc5de36a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GJBKOO5F3NR47AIE4RTFGLN6LM/bundle.json","state_url":"https://pith.science/pith/GJBKOO5F3NR47AIE4RTFGLN6LM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GJBKOO5F3NR47AIE4RTFGLN6LM/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":"2026-06-10T22:28:34Z","links":{"resolver":"https://pith.science/pith/GJBKOO5F3NR47AIE4RTFGLN6LM","bundle":"https://pith.science/pith/GJBKOO5F3NR47AIE4RTFGLN6LM/bundle.json","state":"https://pith.science/pith/GJBKOO5F3NR47AIE4RTFGLN6LM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GJBKOO5F3NR47AIE4RTFGLN6LM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:GJBKOO5F3NR47AIE4RTFGLN6LM","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":"9f8d0cf30ffeb0d90d252cdb21b49693f7b471b0178a835436875f8428a36305","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T15:37:22Z","title_canon_sha256":"923971fb03ffa2bd3cac8f56cf7f7c14510de82f9c13f1c9781ff6c4e514362e"},"schema_version":"1.0","source":{"id":"1303.3810","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3810","created_at":"2026-05-18T00:32:21Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3810v2","created_at":"2026-05-18T00:32:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3810","created_at":"2026-05-18T00:32:21Z"},{"alias_kind":"pith_short_12","alias_value":"GJBKOO5F3NR4","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GJBKOO5F3NR47AIE","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GJBKOO5F","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:46556963d4ce050de21b677231f3a28b3fd75b6df45c2bea9bf6d66fcc5de36a","target":"graph","created_at":"2026-05-18T00:32:21Z","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":"The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set $C$. It is well known that there are compact Hausdorff spaces of cardinality equal to that of $C$ that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of $C$. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.","authors_text":"Fabian Dreher, Tony Samuel","cross_cats":["math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T15:37:22Z","title":"Continuous images of Cantor's ternary set"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3810","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":"sha256:0b54a57f50e239c16d0ba50fb234bedf1d611544b0817b9a07e32feaee4bc465","target":"record","created_at":"2026-05-18T00:32:21Z","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":"9f8d0cf30ffeb0d90d252cdb21b49693f7b471b0178a835436875f8428a36305","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T15:37:22Z","title_canon_sha256":"923971fb03ffa2bd3cac8f56cf7f7c14510de82f9c13f1c9781ff6c4e514362e"},"schema_version":"1.0","source":{"id":"1303.3810","kind":"arxiv","version":2}},"canonical_sha256":"3242a73ba5db63cf8104e466532dbe5b239b3cf2afeb86667161f52016f95490","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3242a73ba5db63cf8104e466532dbe5b239b3cf2afeb86667161f52016f95490","first_computed_at":"2026-05-18T00:32:21.974877Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:21.974877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iXfTgfNmUidDr5e1/vWNDd6J8nJHHazaUN2/SLJxv4Khw7Mvzu69Hej0LVXi619uyIR9QwT/GsLLy8MSHQDpAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:21.975551Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.3810","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b54a57f50e239c16d0ba50fb234bedf1d611544b0817b9a07e32feaee4bc465","sha256:46556963d4ce050de21b677231f3a28b3fd75b6df45c2bea9bf6d66fcc5de36a"],"state_sha256":"aa5bfcd95b5cddf47dc9a8072a6406c7abe92007b3cf0559eb6dc84cc65b1e31"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9UJSKK1lC8T5LjpHk6wkXftJu4wMPVu7oS9Vylrv3acD5OQpwtEpVh4ItHfpfooF2bJYcGKXXpsOwisLsxxfBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T22:28:34.966482Z","bundle_sha256":"cb4e10de987149a41fe9f326f4a166c2537806608eac2e2c12613b895506508d"}}