{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:5VXAGLBSJD5DYI5OQI5QHVI3R2","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":"e1660395115f9687696653437f22b7c86e51d948c1030c1d51bf4b7ea02328e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-20T07:17:16Z","title_canon_sha256":"56774a577579103cbc42b4d9c000f6cd4468b31a8a73781e2f5a37abad8f1b8e"},"schema_version":"1.0","source":{"id":"1309.5190","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5190","created_at":"2026-05-18T03:12:47Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5190v1","created_at":"2026-05-18T03:12:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5190","created_at":"2026-05-18T03:12:47Z"},{"alias_kind":"pith_short_12","alias_value":"5VXAGLBSJD5D","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"5VXAGLBSJD5DYI5O","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"5VXAGLBS","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:1bd7345a6a613a35f50c7ad465338566e2a1dea0e0c5fb94df5d8d9611a3db52","target":"graph","created_at":"2026-05-18T03:12:47Z","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":"Let $X$ be the prime spectrum of a ring. In [arXiv:0707.1525] the authors define a topology on $X$ by using ultrafilters and they show that this topology is precisely the constructible topology. In this paper we generalize the construction given in [arXiv:0707.1525] and, starting from a set $X$ and a collection of subsets $\\mathcal{F}$ of $X$, we define by using ultrafilters a topology on $X$ in which $\\mathcal F$ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several new examples of spectral spaces.","authors_text":"Carmelo A. Finocchiaro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-20T07:17:16Z","title":"Spectral spaces and ultrafilters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5190","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:c71fd7db0aacd7e49c4abb25fcbc5217f5e4c2ed835fb9c47b9b67dc77928c45","target":"record","created_at":"2026-05-18T03:12:47Z","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":"e1660395115f9687696653437f22b7c86e51d948c1030c1d51bf4b7ea02328e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-20T07:17:16Z","title_canon_sha256":"56774a577579103cbc42b4d9c000f6cd4468b31a8a73781e2f5a37abad8f1b8e"},"schema_version":"1.0","source":{"id":"1309.5190","kind":"arxiv","version":1}},"canonical_sha256":"ed6e032c3248fa3c23ae823b03d51b8e8cd7f5c280c56460ac0338c645508d6b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed6e032c3248fa3c23ae823b03d51b8e8cd7f5c280c56460ac0338c645508d6b","first_computed_at":"2026-05-18T03:12:47.341346Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:47.341346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sc0VBfTU6VsCmyWXX/S8gcou+uahjIAVWdkLZ9jfdqS7sJlILfniWj5L03oMwVGwF3OUBsTQpIdFUDpZXFN4CA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:47.341963Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.5190","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c71fd7db0aacd7e49c4abb25fcbc5217f5e4c2ed835fb9c47b9b67dc77928c45","sha256:1bd7345a6a613a35f50c7ad465338566e2a1dea0e0c5fb94df5d8d9611a3db52"],"state_sha256":"e2f7ace6900f47c369c191602379520046609707ac84216d81d28dd7544b55d1"}