{"paper":{"title":"Pinning Down versus Density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.GN","authors_text":"Istv\\'an Juh\\'asz, Lajos Soukup, Zolt\\'an Szentmikl\\'ossy","submitted_at":"2015-05-31T09:06:57Z","abstract_excerpt":"The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $\\kappa$ such that for any neighborhood assignment $U:X\\to \\tau_X$ there is a set $A\\in [X]^\\kappa$ with $A\\cap U(x)\\ne\\emptyset$ for all $x\\in X$. Clearly, c$(X) \\le {pd}(X) \\le {d}(X)$.\n  Here we prove that the following statements are equivalent:\n  (1) $2^\\kappa<\\kappa^{+\\omega}$ for each cardinal $\\kappa$;\n  (2) ${d}(X)={pd}(X)$ for each Hausdorff space $X$;\n  (3) ${d}(X)={pd}(X)$ for each 0-dimensional Hausdorff space $X$.\n  This answers two questions of Banakh and Ravsky.\n  The dispersion character $\\D"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00206","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"}