{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CKL3G5GTB7SVERFNQ4C2UAG6OJ","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":"1a00badb727230ffe20cbe4a3b32bd6a932fa3aab633fabe60adb79f70a57806","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2014-12-03T21:30:11Z","title_canon_sha256":"e6f1edbd77abb2c6cfe3b7a03da9bd9c7389b7ccb06d0468887a94b2595a62d6"},"schema_version":"1.0","source":{"id":"1412.1497","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.1497","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"arxiv_version","alias_value":"1412.1497v1","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.1497","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"pith_short_12","alias_value":"CKL3G5GTB7SV","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CKL3G5GTB7SVERFN","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CKL3G5GT","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:db087db3f26cf2a9bfcc239dd1f4d5087608c2a41adaa03f2c49cfbd06a29206","target":"graph","created_at":"2026-05-18T02:32:07Z","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":"Being motivated by the study of the space $C_c(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with the compact-open topology, we introduced in [15] the concepts of a $cp$-network and a $cn$-network (at a point $x$) in $X$. In the present paper we describe the topology of $X$ admitting a countable $cp$- or $cn$-network at a point $x\\in X$. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [43]. We show that a Baire topological group $G$ is metrizable","authors_text":"J. Kakol, S. S. Gabriyelyan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2014-12-03T21:30:11Z","title":"On topological spaces and topological groups with certain local countable networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1497","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":"sha256:247fc72c6ff5ee8363edcadae87ae71c428eae6dac05941a6ae891819e1edd61","target":"record","created_at":"2026-05-18T02:32:07Z","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":"1a00badb727230ffe20cbe4a3b32bd6a932fa3aab633fabe60adb79f70a57806","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2014-12-03T21:30:11Z","title_canon_sha256":"e6f1edbd77abb2c6cfe3b7a03da9bd9c7389b7ccb06d0468887a94b2595a62d6"},"schema_version":"1.0","source":{"id":"1412.1497","kind":"arxiv","version":1}},"canonical_sha256":"1297b374d30fe55244ad8705aa00de72611a793846aed50d42c68086b5892ef8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1297b374d30fe55244ad8705aa00de72611a793846aed50d42c68086b5892ef8","first_computed_at":"2026-05-18T02:32:07.069904Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:07.069904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JIHvezoC1dxGKSVZZuiRKF5Pq7idbjM16vp2ozK5IhZTVLYAKfLQCQ7VrDi4cHdQXdCE0Wf2mRvzgkmY1Mi7Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:07.070264Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.1497","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:247fc72c6ff5ee8363edcadae87ae71c428eae6dac05941a6ae891819e1edd61","sha256:db087db3f26cf2a9bfcc239dd1f4d5087608c2a41adaa03f2c49cfbd06a29206"],"state_sha256":"3f6d3a06742309f832b08149a6a4702289c594224239a6eee29c54b1b5e8f642"}