{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:P7PQVBCDVXBP2YWEBDDPCQIMQM","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":"31376e8a28848afb6878ab4f5fe3fa8dabd7ea84a20fa59334426b52513b0f0e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-29T11:56:24Z","title_canon_sha256":"837f923b13880c9e1c6bdf53888ed1060c4bdf9fab80b29d7af4c5d5c653f629"},"schema_version":"1.0","source":{"id":"1612.09111","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.09111","created_at":"2026-05-18T00:49:22Z"},{"alias_kind":"arxiv_version","alias_value":"1612.09111v2","created_at":"2026-05-18T00:49:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.09111","created_at":"2026-05-18T00:49:22Z"},{"alias_kind":"pith_short_12","alias_value":"P7PQVBCDVXBP","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"P7PQVBCDVXBP2YWE","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"P7PQVBCD","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:1c513115cc9bf24f36ac324faeb044327e3ad4049d9a8b78a5a45854ddf8a107","target":"graph","created_at":"2026-05-18T00:49:22Z","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":"Recall that a topological space is said to be a $k_\\omega$-space if it is the direct limit of an ascending sequence of compact Hausdorff topological spaces. If each point in a Hausdorff space $X$ has an open neighbourhood which is a $k_\\omega$-space, then $X$ is called locally $k_\\omega$. We show that a topological group is complete whenever the underlying topological space is locally $k_\\omega$. As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete.","authors_text":"Helge Glockner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-29T11:56:24Z","title":"Completeness of locally $k_\\omega$-groups and related infinite-dimensional Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09111","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":"sha256:7fe70fa9f4b7a26d4af2503212c7d46829f01ba9f6d502d96100153fbbcfc787","target":"record","created_at":"2026-05-18T00:49:22Z","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":"31376e8a28848afb6878ab4f5fe3fa8dabd7ea84a20fa59334426b52513b0f0e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-29T11:56:24Z","title_canon_sha256":"837f923b13880c9e1c6bdf53888ed1060c4bdf9fab80b29d7af4c5d5c653f629"},"schema_version":"1.0","source":{"id":"1612.09111","kind":"arxiv","version":2}},"canonical_sha256":"7fdf0a8443adc2fd62c408c6f1410c831707ea17c1a701556c83659daa77911f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7fdf0a8443adc2fd62c408c6f1410c831707ea17c1a701556c83659daa77911f","first_computed_at":"2026-05-18T00:49:22.833911Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:22.833911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s8F5U/EKkJqq1njNDvu1bh82NyOb1+TAT6DhXkG4lL8+qw6n6Tk3Aa6OnQo28TWIqNRUixDFyaczmHDpo3OuAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:22.834383Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.09111","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7fe70fa9f4b7a26d4af2503212c7d46829f01ba9f6d502d96100153fbbcfc787","sha256:1c513115cc9bf24f36ac324faeb044327e3ad4049d9a8b78a5a45854ddf8a107"],"state_sha256":"bfa1c77f1370b7a063f63ad758de62c9271bae19bebcfa72c03a52a1660137a8"}