{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ZL22ORHYFIOBLM3XC5GBHXDLIJ","short_pith_number":"pith:ZL22ORHY","canonical_record":{"source":{"id":"1901.04887","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-01-15T15:47:52Z","cross_cats_sorted":[],"title_canon_sha256":"eb2ee702c53582fd7a2317147e7f003858b7d398c8337e48e0ff50a40944f49c","abstract_canon_sha256":"da392717144c12b2ac1be3229ee54aeac02c0e08c34fa8f7fc552370e9acbf38"},"schema_version":"1.0"},"canonical_sha256":"caf5a744f82a1c15b377174c13dc6b424abfbfc28b1e8ae87e2df75e59a2a29b","source":{"kind":"arxiv","id":"1901.04887","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.04887","created_at":"2026-05-17T23:56:21Z"},{"alias_kind":"arxiv_version","alias_value":"1901.04887v1","created_at":"2026-05-17T23:56:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.04887","created_at":"2026-05-17T23:56:21Z"},{"alias_kind":"pith_short_12","alias_value":"ZL22ORHYFIOB","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"ZL22ORHYFIOBLM3X","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"ZL22ORHY","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ZL22ORHYFIOBLM3XC5GBHXDLIJ","target":"record","payload":{"canonical_record":{"source":{"id":"1901.04887","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-01-15T15:47:52Z","cross_cats_sorted":[],"title_canon_sha256":"eb2ee702c53582fd7a2317147e7f003858b7d398c8337e48e0ff50a40944f49c","abstract_canon_sha256":"da392717144c12b2ac1be3229ee54aeac02c0e08c34fa8f7fc552370e9acbf38"},"schema_version":"1.0"},"canonical_sha256":"caf5a744f82a1c15b377174c13dc6b424abfbfc28b1e8ae87e2df75e59a2a29b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:21.410353Z","signature_b64":"73/xCF7SjrDQ4E53hLIgkSjSzP8WQjLIjXY8d5MReMAdFOXX61pLrfeafL5j7K2UKSHUORU3grGjdG6Y86kuCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"caf5a744f82a1c15b377174c13dc6b424abfbfc28b1e8ae87e2df75e59a2a29b","last_reissued_at":"2026-05-17T23:56:21.409800Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:21.409800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.04887","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":"2026-05-17T23:56:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CqabapTw0LNvjKHvs56wsFdH9l14idskX6n8pCqz9/xOC+T2MqqfKK1N7md3YBtJUCf1hu2AAtm7H5l5Pu5qCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T20:13:48.285813Z"},"content_sha256":"f8eab10ddb84294eddba14d8a7c7c5cf8834486843becce55dfac81b09d3cae3","schema_version":"1.0","event_id":"sha256:f8eab10ddb84294eddba14d8a7c7c5cf8834486843becce55dfac81b09d3cae3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ZL22ORHYFIOBLM3XC5GBHXDLIJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On weakening tightness to weak tightness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Angelo Bella, Nathan Carlson","submitted_at":"2019-01-15T15:47:52Z","abstract_excerpt":"The weak tightness $wt(X)$ of a space $X$ was introduced in [11] with the property $wt(X)\\leq t(X)$. We investigate several well-known results concerning $t(X)$ and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum $X$ such that $wt(X)=\\aleph_0<t(X)$ under $2^{\\aleph_0}=2^{\\aleph_1}$. In particular, this demonstrates the celebrated Balogh's Theorem [5] does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if $X$ is a homogeneous com"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04887","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":"2026-05-17T23:56:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tBHOv1IT7df7ONg4C+Qk5z4zgL8TfFFV8kWMDFla9TqSUfrjPFpRYN3AIME7m9L7lZKiYNoVPZUA77qE2K1JAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T20:13:48.286168Z"},"content_sha256":"43c1dc5f1575b55d5bda3e57bf2062a41eab65dcbb837f9ab97662261ccbf45f","schema_version":"1.0","event_id":"sha256:43c1dc5f1575b55d5bda3e57bf2062a41eab65dcbb837f9ab97662261ccbf45f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZL22ORHYFIOBLM3XC5GBHXDLIJ/bundle.json","state_url":"https://pith.science/pith/ZL22ORHYFIOBLM3XC5GBHXDLIJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZL22ORHYFIOBLM3XC5GBHXDLIJ/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":"2026-05-31T20:13:48Z","links":{"resolver":"https://pith.science/pith/ZL22ORHYFIOBLM3XC5GBHXDLIJ","bundle":"https://pith.science/pith/ZL22ORHYFIOBLM3XC5GBHXDLIJ/bundle.json","state":"https://pith.science/pith/ZL22ORHYFIOBLM3XC5GBHXDLIJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZL22ORHYFIOBLM3XC5GBHXDLIJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZL22ORHYFIOBLM3XC5GBHXDLIJ","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":"da392717144c12b2ac1be3229ee54aeac02c0e08c34fa8f7fc552370e9acbf38","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-01-15T15:47:52Z","title_canon_sha256":"eb2ee702c53582fd7a2317147e7f003858b7d398c8337e48e0ff50a40944f49c"},"schema_version":"1.0","source":{"id":"1901.04887","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.04887","created_at":"2026-05-17T23:56:21Z"},{"alias_kind":"arxiv_version","alias_value":"1901.04887v1","created_at":"2026-05-17T23:56:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.04887","created_at":"2026-05-17T23:56:21Z"},{"alias_kind":"pith_short_12","alias_value":"ZL22ORHYFIOB","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"ZL22ORHYFIOBLM3X","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"ZL22ORHY","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:43c1dc5f1575b55d5bda3e57bf2062a41eab65dcbb837f9ab97662261ccbf45f","target":"graph","created_at":"2026-05-17T23:56:21Z","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 weak tightness $wt(X)$ of a space $X$ was introduced in [11] with the property $wt(X)\\leq t(X)$. We investigate several well-known results concerning $t(X)$ and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum $X$ such that $wt(X)=\\aleph_0<t(X)$ under $2^{\\aleph_0}=2^{\\aleph_1}$. In particular, this demonstrates the celebrated Balogh's Theorem [5] does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if $X$ is a homogeneous com","authors_text":"Angelo Bella, Nathan Carlson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-01-15T15:47:52Z","title":"On weakening tightness to weak tightness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04887","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:f8eab10ddb84294eddba14d8a7c7c5cf8834486843becce55dfac81b09d3cae3","target":"record","created_at":"2026-05-17T23:56:21Z","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":"da392717144c12b2ac1be3229ee54aeac02c0e08c34fa8f7fc552370e9acbf38","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-01-15T15:47:52Z","title_canon_sha256":"eb2ee702c53582fd7a2317147e7f003858b7d398c8337e48e0ff50a40944f49c"},"schema_version":"1.0","source":{"id":"1901.04887","kind":"arxiv","version":1}},"canonical_sha256":"caf5a744f82a1c15b377174c13dc6b424abfbfc28b1e8ae87e2df75e59a2a29b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"caf5a744f82a1c15b377174c13dc6b424abfbfc28b1e8ae87e2df75e59a2a29b","first_computed_at":"2026-05-17T23:56:21.409800Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:21.409800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"73/xCF7SjrDQ4E53hLIgkSjSzP8WQjLIjXY8d5MReMAdFOXX61pLrfeafL5j7K2UKSHUORU3grGjdG6Y86kuCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:21.410353Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.04887","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f8eab10ddb84294eddba14d8a7c7c5cf8834486843becce55dfac81b09d3cae3","sha256:43c1dc5f1575b55d5bda3e57bf2062a41eab65dcbb837f9ab97662261ccbf45f"],"state_sha256":"676b2017ab3b12e7b791185012ac01d767242c40a921fe0b53dbdfc226a855d9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mtMH49G7up3OmSONgDfvLbdByQNsy+YrRInsMee61aj7hCcOvQcNWMP0uyTa8FoocgZQRqH0v62puUpXIiQKCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T20:13:48.288853Z","bundle_sha256":"f3d64904c7ae7e4dd72209e1bfd1ab2f3461c2995c9e0e4bbab019817d895e1f"}}