{"paper":{"title":"On open-open games of uncountable length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Andrzej Kucharski","submitted_at":"2012-08-06T15:03:28Z","abstract_excerpt":"The aim of this note is to investigate the open-open game of uncountable length. We introduce a cardinal number $\\mu(X)$, which says how long the Player I has to play to ensure a victory. It is proved that $\\su(X)\\leq\\mu(X)\\leq\\su(X)^+$. We also introduce the class $\\mathcal C_\\kappa$ of topological spaces that can be represented as the inverse limit of $\\kappa$-complete system $\\{X_\\sigma,\\pi^\\sigma_\\rho,\\Sigma\\}$ with $\\w(X_\\sigma)\\leq\\kappa$ and skeletal bonding maps. It is shown that product of spaces which belong to $\\mathcal C_\\kappa$ also belongs to this class and $\\mu(X)\\leq\\kappa$ whe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1182","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"}