{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MTA3IXVOAUVO45VBISCMSTCF44","short_pith_number":"pith:MTA3IXVO","canonical_record":{"source":{"id":"1605.05630","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-05-18T15:51:36Z","cross_cats_sorted":[],"title_canon_sha256":"3de48538f1cc16c919c59ac734016085c18b78fdd5bfa11b81c8f8a5f14e4d90","abstract_canon_sha256":"7a0339aea9d2ed144cf391bcdf6b5c1a5702afcb9bca61d4b8da61dba8df1435"},"schema_version":"1.0"},"canonical_sha256":"64c1b45eae052aee76a14484c94c45e70d47573c6a60135fb0580fc7b3679590","source":{"kind":"arxiv","id":"1605.05630","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.05630","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"arxiv_version","alias_value":"1605.05630v2","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.05630","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"pith_short_12","alias_value":"MTA3IXVOAUVO","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MTA3IXVOAUVO45VB","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MTA3IXVO","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MTA3IXVOAUVO45VBISCMSTCF44","target":"record","payload":{"canonical_record":{"source":{"id":"1605.05630","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-05-18T15:51:36Z","cross_cats_sorted":[],"title_canon_sha256":"3de48538f1cc16c919c59ac734016085c18b78fdd5bfa11b81c8f8a5f14e4d90","abstract_canon_sha256":"7a0339aea9d2ed144cf391bcdf6b5c1a5702afcb9bca61d4b8da61dba8df1435"},"schema_version":"1.0"},"canonical_sha256":"64c1b45eae052aee76a14484c94c45e70d47573c6a60135fb0580fc7b3679590","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:13.922013Z","signature_b64":"NHWdIIeU8jt5QpsFkl1fTb69wfrNvKQy2wxigFGUzYpzdSWKSSIy8evSJvhCiBsNKwStnKGZAJ01vV4+Z9TkDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64c1b45eae052aee76a14484c94c45e70d47573c6a60135fb0580fc7b3679590","last_reissued_at":"2026-05-18T00:40:13.921291Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:13.921291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.05630","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":"2026-05-18T00:40:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vJ5Y54gDXFhq6R4aUIFDMtrdoRL8G1vFDM5ZfFMVZsB9jCzQHz5lQ37EuDgLNsvoIH9By//B96y3ElAxD97CBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T20:29:04.405092Z"},"content_sha256":"1ac973fa0ee09740cf359d907b941d7b17619ed376a221819f4392fce1e83413","schema_version":"1.0","event_id":"sha256:1ac973fa0ee09740cf359d907b941d7b17619ed376a221819f4392fce1e83413"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MTA3IXVOAUVO45VBISCMSTCF44","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$G_\\delta$ covers of compact spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Paul Szeptycki, Santi Spadaro","submitted_at":"2016-05-18T15:51:36Z","abstract_excerpt":"We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_\\delta$ cover with no continuum-sized ($G_\\delta$)-dense subcollection. We also prove that in a countably compact weakly Lindel\\\"of normal space of countable tightness, every $G_\\delta$ cover has a $\\mathfrak{c}$-sized subcollection with a $G_\\delta$-dense union and that in a Lindel\\\"of space with a base of multiplicity continuum, every $G_\\delta$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05630","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":"2026-05-18T00:40:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9DvcbtDeaSOetX7eTKQ2V5D+z9FCluzjuix/QxkWa1t/n/f7BpWHhqk9YZ/16WKE6qj/pZn6oAeXokPca5iBDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T20:29:04.405455Z"},"content_sha256":"027dd162d6af732dfcc60621d2ec017e7210a8a7fd45c5dfeba748f1e5badaa7","schema_version":"1.0","event_id":"sha256:027dd162d6af732dfcc60621d2ec017e7210a8a7fd45c5dfeba748f1e5badaa7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTA3IXVOAUVO45VBISCMSTCF44/bundle.json","state_url":"https://pith.science/pith/MTA3IXVOAUVO45VBISCMSTCF44/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTA3IXVOAUVO45VBISCMSTCF44/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":"2026-06-10T20:29:04Z","links":{"resolver":"https://pith.science/pith/MTA3IXVOAUVO45VBISCMSTCF44","bundle":"https://pith.science/pith/MTA3IXVOAUVO45VBISCMSTCF44/bundle.json","state":"https://pith.science/pith/MTA3IXVOAUVO45VBISCMSTCF44/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTA3IXVOAUVO45VBISCMSTCF44/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MTA3IXVOAUVO45VBISCMSTCF44","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":"7a0339aea9d2ed144cf391bcdf6b5c1a5702afcb9bca61d4b8da61dba8df1435","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-05-18T15:51:36Z","title_canon_sha256":"3de48538f1cc16c919c59ac734016085c18b78fdd5bfa11b81c8f8a5f14e4d90"},"schema_version":"1.0","source":{"id":"1605.05630","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.05630","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"arxiv_version","alias_value":"1605.05630v2","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.05630","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"pith_short_12","alias_value":"MTA3IXVOAUVO","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MTA3IXVOAUVO45VB","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MTA3IXVO","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:027dd162d6af732dfcc60621d2ec017e7210a8a7fd45c5dfeba748f1e5badaa7","target":"graph","created_at":"2026-05-18T00:40:13Z","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":"We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_\\delta$ cover with no continuum-sized ($G_\\delta$)-dense subcollection. We also prove that in a countably compact weakly Lindel\\\"of normal space of countable tightness, every $G_\\delta$ cover has a $\\mathfrak{c}$-sized subcollection with a $G_\\delta$-dense union and that in a Lindel\\\"of space with a base of multiplicity continuum, every $G_\\delta$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Ve","authors_text":"Paul Szeptycki, Santi Spadaro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-05-18T15:51:36Z","title":"$G_\\delta$ covers of compact spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05630","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":"sha256:1ac973fa0ee09740cf359d907b941d7b17619ed376a221819f4392fce1e83413","target":"record","created_at":"2026-05-18T00:40:13Z","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":"7a0339aea9d2ed144cf391bcdf6b5c1a5702afcb9bca61d4b8da61dba8df1435","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-05-18T15:51:36Z","title_canon_sha256":"3de48538f1cc16c919c59ac734016085c18b78fdd5bfa11b81c8f8a5f14e4d90"},"schema_version":"1.0","source":{"id":"1605.05630","kind":"arxiv","version":2}},"canonical_sha256":"64c1b45eae052aee76a14484c94c45e70d47573c6a60135fb0580fc7b3679590","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64c1b45eae052aee76a14484c94c45e70d47573c6a60135fb0580fc7b3679590","first_computed_at":"2026-05-18T00:40:13.921291Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:13.921291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NHWdIIeU8jt5QpsFkl1fTb69wfrNvKQy2wxigFGUzYpzdSWKSSIy8evSJvhCiBsNKwStnKGZAJ01vV4+Z9TkDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:13.922013Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.05630","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1ac973fa0ee09740cf359d907b941d7b17619ed376a221819f4392fce1e83413","sha256:027dd162d6af732dfcc60621d2ec017e7210a8a7fd45c5dfeba748f1e5badaa7"],"state_sha256":"eddd93a0292320d3cfa85394a39576d3cbbf5ca1491a8c57fb2df7a586664278"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FMOdPmzaltOmEmesrTTP3Gkeigqg3LhRmlwXeqkYnTXz6xZzpVVIAVjbVOHYKVKtRTbhRlWsizOBp4/seYF7Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T20:29:04.407464Z","bundle_sha256":"536d865142b75c9db3819cc2c87debda7e4a02d258a0d706722d9bcb62196b33"}}