{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7FWPJE5ZEZFJO7ALX5YQX7DVEB","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":"9981aa2363e66e79943f4155cfd4b8a8caafbdc1166a8c8cf9e022d42d580f0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-25T19:15:21Z","title_canon_sha256":"a19564acbf26018736b93a825305e7e337be3a79fea26dbd0237921cadaf7f29"},"schema_version":"1.0","source":{"id":"1510.07291","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07291","created_at":"2026-05-18T01:27:14Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07291v2","created_at":"2026-05-18T01:27:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07291","created_at":"2026-05-18T01:27:14Z"},{"alias_kind":"pith_short_12","alias_value":"7FWPJE5ZEZFJ","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7FWPJE5ZEZFJO7AL","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7FWPJE5Z","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:ac60bb30874d9cb5932b37bc0f022c454dce2cb2491aadc12cdb808bf605288e","target":"graph","created_at":"2026-05-18T01:27:14Z","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 study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably homeomorphic to a definable set equipped with its euclidean topology. We show that almost every point in a definable metric space has a neighborhood which is definably homeomorphic to an open definable su","authors_text":"Erik Walsberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-25T19:15:21Z","title":"On the Topology of Metric Spaces definable in o-minimal expansions of fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07291","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":"sha256:626610d34612a65615e62db7ef6a2065ec35a469bfa8fb7e5189bc833eda500b","target":"record","created_at":"2026-05-18T01:27:14Z","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":"9981aa2363e66e79943f4155cfd4b8a8caafbdc1166a8c8cf9e022d42d580f0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-25T19:15:21Z","title_canon_sha256":"a19564acbf26018736b93a825305e7e337be3a79fea26dbd0237921cadaf7f29"},"schema_version":"1.0","source":{"id":"1510.07291","kind":"arxiv","version":2}},"canonical_sha256":"f96cf493b9264a977c0bbf710bfc752078ec4c6f3a04fc4826bfa8efdab0aa71","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f96cf493b9264a977c0bbf710bfc752078ec4c6f3a04fc4826bfa8efdab0aa71","first_computed_at":"2026-05-18T01:27:14.800228Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:27:14.800228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Qal7NnQwNAsNiiDsjuzHXuAdP2pZ4Dvw8UrYIVSfcoiNwxOCQWvz7PBT7JGcJXVD04zRchzBEg/5cU+8zttDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:27:14.800843Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07291","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:626610d34612a65615e62db7ef6a2065ec35a469bfa8fb7e5189bc833eda500b","sha256:ac60bb30874d9cb5932b37bc0f022c454dce2cb2491aadc12cdb808bf605288e"],"state_sha256":"9760b01b438db36f967aaef70e7af283f7de6a7eed8902765709f75edf129f9a"}