{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:V66IQIW6RVUPIK3LWE2WTHUIGA","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":"e9353d4b5c686bacfa734245d9642e29e7eb0d4fbd85b75e29bcb40c7111e360","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-04-30T09:04:26Z","title_canon_sha256":"94c2316e1c930476e7645fc0a35e3211bf02301802f784128feb4dc1c0b3d3d2"},"schema_version":"1.0","source":{"id":"1504.08130","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.08130","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"arxiv_version","alias_value":"1504.08130v1","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.08130","created_at":"2026-05-18T02:17:23Z"},{"alias_kind":"pith_short_12","alias_value":"V66IQIW6RVUP","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"V66IQIW6RVUPIK3L","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"V66IQIW6","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:d244af4d6b07752e1349b9ec2487c16534e1bc4fcdbab40b7f0849252012a317","target":"graph","created_at":"2026-05-18T02:17:23Z","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":"Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpi\\'nski and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric $\\sigma$-discrete spaces. Some related topics are also explored. For example: For each infinite cardinal number $\\frak m$, there exist $2^{\\frak m}$ many non-homeomorphic metric scattered spaces of the cardinality $\\frak m $; If $X \\subseteq \\omega_1$ is a stationary set, then the poset formed from dimensional types of subspaces of $X$ contains uncountable a","authors_text":"Marta Walczy\\'nska, Szymon Plewik","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-04-30T09:04:26Z","title":"Embeddable properties of metric $\\sigma$-discrete spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08130","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":"sha256:81222a69a6efe350fe80bb09dab926d49fe86bf4b673424ca95ae6c0ad59f207","target":"record","created_at":"2026-05-18T02:17:23Z","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":"e9353d4b5c686bacfa734245d9642e29e7eb0d4fbd85b75e29bcb40c7111e360","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-04-30T09:04:26Z","title_canon_sha256":"94c2316e1c930476e7645fc0a35e3211bf02301802f784128feb4dc1c0b3d3d2"},"schema_version":"1.0","source":{"id":"1504.08130","kind":"arxiv","version":1}},"canonical_sha256":"afbc8822de8d68f42b6bb135699e8830384f1c7d04bfb6a637d142e301d71d08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"afbc8822de8d68f42b6bb135699e8830384f1c7d04bfb6a637d142e301d71d08","first_computed_at":"2026-05-18T02:17:23.242681Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:23.242681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sLzogljO1NsVfzYz8OvKPaSG8CIcowFflF8dMEQ/06lBPmStx6NcrW+zDCiWTsC0GNWkAya0x2ZPjyv2HTr+CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:23.243413Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.08130","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:81222a69a6efe350fe80bb09dab926d49fe86bf4b673424ca95ae6c0ad59f207","sha256:d244af4d6b07752e1349b9ec2487c16534e1bc4fcdbab40b7f0849252012a317"],"state_sha256":"c0c5d72fb116853abd693fdaeb640e00a53632498fe83cc64ba779d84c329d03"}