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We prove that $\\diamondsuit$ implies that there is an $HFC_w$ space in $2^{\\omega_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindel\\\"of) which is not strongly $D$. We also show that any $HFC$ space $X$ is dually discrete and if additionally, countable sets have Menger closure then $X$ is a $D$-space."},"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":"1902.06500","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-02-18T10:38:51Z","cross_cats_sorted":[],"title_canon_sha256":"f271cc60ea52a228c669b8722e9cfff6e75f7efb7a33cce48cd5b5c61c108b3a","abstract_canon_sha256":"3106f29d8696b25ca07f887f851fe4071335f2133e1272bc8362a8ac06e481c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:45.269344Z","signature_b64":"KgRqAnZcqSn23MVm00mjiC2iyOn1YYsn6qhK6MHMyA+AAL6DRqVRy7kuKGupWs+/bpyfGYWgc3Xd+ZkHrvpnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49ad6a70cca8e84bb2d7854d74a3e00f5bda5bf52fdac7f9b23bad49adae2600","last_reissued_at":"2026-05-17T23:53:45.268780Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:45.268780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A 0-dimensional, Lindel\\\"of space that is not strongly D","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Daniel T. Soukup, Paul J. Szeptycki","submitted_at":"2019-02-18T10:38:51Z","abstract_excerpt":"A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\\{U_x:x\\in X\\}$, there is a $D\\subseteq X$ such that $\\{U_x:x\\in D\\}$ covers $X$ and $D$ is locally finite in the topology generated by $\\{U_x:x\\in X\\}$. We prove that $\\diamondsuit$ implies that there is an $HFC_w$ space in $2^{\\omega_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindel\\\"of) which is not strongly $D$. We also show that any $HFC$ space $X$ is dually discrete and if additionally, countable sets have Menger closure then $X$ is a $D$-space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.06500","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":"1902.06500","created_at":"2026-05-17T23:53:45.268873+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.06500v1","created_at":"2026-05-17T23:53:45.268873+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.06500","created_at":"2026-05-17T23:53:45.268873+00:00"},{"alias_kind":"pith_short_12","alias_value":"JGWWU4GMVDUE","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"JGWWU4GMVDUEXMWX","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"JGWWU4GM","created_at":"2026-05-18T12:33:21.387695+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/JGWWU4GMVDUEXMWXQVGXJI7AB5","json":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5.json","graph_json":"https://pith.science/api/pith-number/JGWWU4GMVDUEXMWXQVGXJI7AB5/graph.json","events_json":"https://pith.science/api/pith-number/JGWWU4GMVDUEXMWXQVGXJI7AB5/events.json","paper":"https://pith.science/paper/JGWWU4GM"},"agent_actions":{"view_html":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5","download_json":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5.json","view_paper":"https://pith.science/paper/JGWWU4GM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.06500&json=true","fetch_graph":"https://pith.science/api/pith-number/JGWWU4GMVDUEXMWXQVGXJI7AB5/graph.json","fetch_events":"https://pith.science/api/pith-number/JGWWU4GMVDUEXMWXQVGXJI7AB5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5/action/storage_attestation","attest_author":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5/action/author_attestation","sign_citation":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5/action/citation_signature","submit_replication":"https://pith.science/pith/JGWWU4GMVDUEXMWXQVGXJI7AB5/action/replication_record"}},"created_at":"2026-05-17T23:53:45.268873+00:00","updated_at":"2026-05-17T23:53:45.268873+00:00"}