{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:LACBWOOKVYL6EW6TGQC6U544VR","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":"c09bbca65c22323f77f648faa37b57bfeb2b277143d3c8416c2f5510fed799bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-12-22T20:57:41Z","title_canon_sha256":"9abe814d150e08e6a6f47665e78b85b27b72552f6ecd62e51eb008aea126c84a"},"schema_version":"1.0","source":{"id":"1212.5724","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.5724","created_at":"2026-05-18T03:32:58Z"},{"alias_kind":"arxiv_version","alias_value":"1212.5724v2","created_at":"2026-05-18T03:32:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5724","created_at":"2026-05-18T03:32:58Z"},{"alias_kind":"pith_short_12","alias_value":"LACBWOOKVYL6","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"LACBWOOKVYL6EW6T","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"LACBWOOK","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:8bd3b713dd2e7694df5ba1339d9737110c0b99752f4fe222289384565dee52e0","target":"graph","created_at":"2026-05-18T03:32:58Z","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":"Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindel\\\"of first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic version of cellularity, a special case of which has been introduced by Aurichi. We obtain a game-theoretic proof of Shapirovskii's bound for the number of regular op","authors_text":"Angelo Bella, Santi Spadaro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-12-22T20:57:41Z","title":"Infinite games and cardinal properties of topological spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5724","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":"sha256:12198ccab025b3953d420f12a9e7deef530c11691e7713f5ef9d3c71766c2a3f","target":"record","created_at":"2026-05-18T03:32:58Z","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":"c09bbca65c22323f77f648faa37b57bfeb2b277143d3c8416c2f5510fed799bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-12-22T20:57:41Z","title_canon_sha256":"9abe814d150e08e6a6f47665e78b85b27b72552f6ecd62e51eb008aea126c84a"},"schema_version":"1.0","source":{"id":"1212.5724","kind":"arxiv","version":2}},"canonical_sha256":"58041b39caae17e25bd33405ea779cac684bd86848bb302494c97fa7b0e807ca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58041b39caae17e25bd33405ea779cac684bd86848bb302494c97fa7b0e807ca","first_computed_at":"2026-05-18T03:32:58.475460Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:58.475460Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rAoAQ7UJFsGdi98eZKUurduPjMYHBdQjoFoSDh3nCCTx4kkUpBz/C0eX0zOmYUkV5dwRUURWM6frFBPwYfaHAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:58.476104Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.5724","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:12198ccab025b3953d420f12a9e7deef530c11691e7713f5ef9d3c71766c2a3f","sha256:8bd3b713dd2e7694df5ba1339d9737110c0b99752f4fe222289384565dee52e0"],"state_sha256":"efffafb7f4a54702801ab830814a025df0ea24474aed4c6b605cbc6663db2795"}