{"paper":{"title":"Perfect subsets of generalized Baire spaces and long games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Philipp Schlicht","submitted_at":"2017-03-29T17:27:02Z","abstract_excerpt":"We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space ${}^\\lambda\\lambda$, where $\\lambda$ is an uncountable cardinal with $\\lambda^{<\\lambda}=\\lambda$. In the first main theorem, we show that that the perfect set property for all subsets of ${}^{\\lambda}\\lambda$ that are definable from elements of ${}^\\lambda\\mathrm{Ord}$ is consistent relative to the existence of an inaccessible cardinal above $\\lambda$. In the second main theorem, we introduce a Banach-Mazur type game of length $\\lambda$ and show that the determinacy of this game, for all subs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10148","kind":"arxiv","version":2},"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"}