{"paper":{"title":"Uniquely Universal Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Arnold W. Miller","submitted_at":"2011-06-08T19:31:33Z","abstract_excerpt":"We say that X x Y satisfies the Uniquely Universal property (UU) iff there exists a set U open in X x Y such that for every open set W in Y there is a unique cross section U_x of U with U_x=W. Michael Hrusak raised the question of when does X x Y satisfy UU and noted that if Y is compact then X must have an isolated point.\n  We prove the following:\n  1. If Y is a locally compact noncompact Polish space,\n  then C x Y has UU where C is the Cantor space.\n  2. If Y is Polish, then B x Y has UU iff Y is not compact\n  where B is the Baire space.\n  3. If Y is a sigma-compact subset of a Polish space "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.1629","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"}