{"paper":{"title":"The $\\pi$-property of a Banach space along a filter","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.FA","authors_text":"Jaros{\\l}aw Swaczyna, Tomasz Kania","submitted_at":"2025-08-28T22:13:11Z","abstract_excerpt":"We examine the analyticity of the class of separable Banach spaces possessing the $\\pi$-property, defined in terms of convergence along a filter. Our results establish that this class is $\\Sigma^1_3$ whenever the underlying filter is analytic (as a subset of the Cantor set $\\Delta$). Furthermore, we demonstrate that if the filter is countably generated, the class of such spaces is $\\Sigma^1_2$ with respect to any admissible Polish topology on the family of closed subspaces of $C(\\Delta)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.21242","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.21242/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}