{"paper":{"title":"Quasi-Polish spaces and spaces of filters in second-order arithmetic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Quasi-Polish spaces have equivalent representations as UF spaces, NP spaces, Π₂⁰ subspaces of P(N), and sober spaces of countably presented frames, all formalizable in second-order arithmetic.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Keita Yokoyama, Yuzuki Kaneko","submitted_at":"2026-05-14T16:45:57Z","abstract_excerpt":"The class of quasi-Polish spaces admits several equivalent representations, including UF spaces, NP spaces, $\\mathbf{\\Pi}_2^0$ subspaces of $\\mathcal{P}(\\mathbb{N})$, and sober spaces of countably presented frames. In this paper, we formalize these structures within second-order arithmetic and conduct a systematic reverse mathematical analysis of the transitions between them."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The class of quasi-Polish spaces admits several equivalent representations, including UF spaces, NP spaces, Π₂⁰ subspaces of P(N), and sober spaces of countably presented frames; these structures are formalized in second-order arithmetic and the transitions between them receive a systematic reverse-mathematical analysis.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the listed representations remain equivalent when interpreted inside the language and axioms of second-order arithmetic, without requiring extra set-existence principles beyond those already present in the base theory.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Quasi-Polish spaces and their equivalent representations are formalized in second-order arithmetic, with reverse-mathematical analysis of the transitions between them.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Quasi-Polish spaces have equivalent representations as UF spaces, NP spaces, Π₂⁰ subspaces of P(N), and sober spaces of countably presented frames, all formalizable in second-order arithmetic.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"177a1842a657af29edbf6418fd7d72c42d8154ef683423df6be28b2b0f9f245f"},"source":{"id":"2605.15052","kind":"arxiv","version":1},"verdict":{"id":"a6ada94e-48e9-473e-be6c-3373c2bd3d05","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:11:27.499677Z","strongest_claim":"The class of quasi-Polish spaces admits several equivalent representations, including UF spaces, NP spaces, Π₂⁰ subspaces of P(N), and sober spaces of countably presented frames; these structures are formalized in second-order arithmetic and the transitions between them receive a systematic reverse-mathematical analysis.","one_line_summary":"Quasi-Polish spaces and their equivalent representations are formalized in second-order arithmetic, with reverse-mathematical analysis of the transitions between them.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the listed representations remain equivalent when interpreted inside the language and axioms of second-order arithmetic, without requiring extra set-existence principles beyond those already present in the base theory.","pith_extraction_headline":"Quasi-Polish spaces have equivalent representations as UF spaces, NP spaces, Π₂⁰ subspaces of P(N), and sober spaces of countably presented frames, all formalizable in second-order arithmetic."},"references":{"count":11,"sample":[{"doi":"","year":1987,"title":"Notions of closed subsets of a complete separa ble metric space in weak subsystems of second-order arithmetic","work_id":"90dc8b33-9b06-4ffe-befb-64da8f519af9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"Quasi-Polish spaces","work_id":"1444823c-b353-4f6c-b077-69a7b063a74f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"A generalization of a theorem of hurewicz fo r quasi- polish spaces","work_id":"5dbc31c5-e8fb-43d4-a52c-9dab64df06e5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Ide al pre- sentations and numberings of some classes of effective quasi-Polish spaces","work_id":"8a305d42-624a-4f1b-a1bd-02eb0286bc13","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Spatiality of countably presentable locales (p roved with the baire category theorem)","work_id":"b1ab17f0-92a7-40bb-967d-f153704a8e48","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"f2855e1a99641ce05a6f96ba9eb9ce5306b87ba8f17159a4b96757e38e1ef85e","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d5b8bbf45326ad294ec30a658cc56ef1fc453632595d89496f008512c5d2f60d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}