{"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"}