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The lattice of $\\mathfrak O$-valued functions of $t\\in(0,\\infty)$ with the point-wise partial order and convergence contains the family ${I\\mathfrak O}=\\{A(\\cdot)\\,|\\,\\,A(t)=A^t,\\,\\,A\\in{\\mathfrak O}\\}$. Let $\\widetilde\\Omega$ be the set of atoms of the order closure $\\overline{I\\mathfrak O}$. We describe a class of spaces for which the set $\\widetilde\\Omega$, equ"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.04317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-01-10T21:22:08Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"6850a64a893317c1466774448b6e5ced83991d5674ff2e1e34b57f87edb5ca28","abstract_canon_sha256":"b3e97539e78eadf02d5e654bff4d1328567ccdd2daf6d7b2ac95a09cfd2ae085"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:25.850138Z","signature_b64":"wydyQlbaWVH22uf0UVZyT0Auv3FE2JZQGrHxPOWKuEVdNcLpIl0Pkg3SUYqboocKGTAsDmkDO5BSLWhB6OL+Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28980f9f801504b077f0485c7bddbacb8eb568ea3f68cf9f416a7dd52b13883c","last_reissued_at":"2026-05-17T23:56:25.849715Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:25.849715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The wave model of metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.FA","authors_text":"M. I. Belishev, S. A. Simonov","submitted_at":"2019-01-10T21:22:08Z","abstract_excerpt":"Let $\\Omega$ be a metric space, $A^t$ denote the metric neighborhood of the set $A\\subset\\Omega$ of the radius $t$; ${\\mathfrak O}$ be the lattice of open sets in $\\Omega$ with the partial order $\\subseteq$ and the order convergence. The lattice of $\\mathfrak O$-valued functions of $t\\in(0,\\infty)$ with the point-wise partial order and convergence contains the family ${I\\mathfrak O}=\\{A(\\cdot)\\,|\\,\\,A(t)=A^t,\\,\\,A\\in{\\mathfrak O}\\}$. Let $\\widetilde\\Omega$ be the set of atoms of the order closure $\\overline{I\\mathfrak O}$. We describe a class of spaces for which the set $\\widetilde\\Omega$, equ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04317","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.04317","created_at":"2026-05-17T23:56:25.849780+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.04317v1","created_at":"2026-05-17T23:56:25.849780+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.04317","created_at":"2026-05-17T23:56:25.849780+00:00"},{"alias_kind":"pith_short_12","alias_value":"FCMA7H4ACUCL","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_16","alias_value":"FCMA7H4ACUCLA57Q","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_8","alias_value":"FCMA7H4A","created_at":"2026-05-18T12:33:15.570797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO","json":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO.json","graph_json":"https://pith.science/api/pith-number/FCMA7H4ACUCLA57QJBOHXXN2ZO/graph.json","events_json":"https://pith.science/api/pith-number/FCMA7H4ACUCLA57QJBOHXXN2ZO/events.json","paper":"https://pith.science/paper/FCMA7H4A"},"agent_actions":{"view_html":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO","download_json":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO.json","view_paper":"https://pith.science/paper/FCMA7H4A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.04317&json=true","fetch_graph":"https://pith.science/api/pith-number/FCMA7H4ACUCLA57QJBOHXXN2ZO/graph.json","fetch_events":"https://pith.science/api/pith-number/FCMA7H4ACUCLA57QJBOHXXN2ZO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO/action/storage_attestation","attest_author":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO/action/author_attestation","sign_citation":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO/action/citation_signature","submit_replication":"https://pith.science/pith/FCMA7H4ACUCLA57QJBOHXXN2ZO/action/replication_record"}},"created_at":"2026-05-17T23:56:25.849780+00:00","updated_at":"2026-05-17T23:56:25.849780+00:00"}