{"paper":{"title":"O-minimal open core is not an elementary property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Having an o-minimal open core is not an elementary property.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Alexi Block Gorman, Esther Elbaz Saban","submitted_at":"2026-05-13T15:40:59Z","abstract_excerpt":"Given a structure $\\mathcal{M}$ with a definable topology, its open core is a structure defined on the same universe whose language consists of all open sets of all arities definable in $\\mathcal{M}$. In response to questions raised by Dolich, Miller, and Steinhorn in their early work on open core, we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure $(\\mathbb{Q},<)$ that has an o-minimal open core, but some of its elementary superstructures do not."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure (Q,<) that has an o-minimal open core, but some of its elementary superstructures do not.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the specific expansion of (Q,<) can be chosen so its open core is o-minimal while some elementary extension has a non-o-minimal open core; this relies on the construction preserving the necessary first-order properties.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"O-minimality of the open core is not an elementary property, shown via a counterexample expansion of (Q, <) whose elementary superstructures can lack the property.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Having an o-minimal open core is not an elementary property.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"02d65a4522b694389957bb75f36d469ea1de43da45469d6b92b2ed13cff66ab6"},"source":{"id":"2605.13683","kind":"arxiv","version":1},"verdict":{"id":"5f6869f5-ddd1-4c85-a085-b75adf9303ff","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:04:46.125770Z","strongest_claim":"we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure (Q,<) that has an o-minimal open core, but some of its elementary superstructures do not.","one_line_summary":"O-minimality of the open core is not an elementary property, shown via a counterexample expansion of (Q, <) whose elementary superstructures can lack the property.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the specific expansion of (Q,<) can be chosen so its open core is o-minimal while some elementary extension has a non-o-minimal open core; this relies on the construction preserving the necessary first-order properties.","pith_extraction_headline":"Having an o-minimal open core is not an elementary property."},"references":{"count":9,"sample":[{"doi":"","year":2020,"title":"Alexi Block Gorman, Philipp Hieronymi, and Elliot Kaplan, Pairs of Theories Satisfying a Mordell-Lang Condition. Fund. Math., (2) 251:131-160, (2020)","work_id":"235c927d-d80f-4ae4-88b0-ce2c19f73f09","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Gareth Boxall and Philipp Hieronymi, Expansions which introduce no new open sets.J. Symb. Log., 77(1):111-121, (2012)","work_id":"c3356608-a2e0-4f33-928a-34e4e822b62f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Alfred Dolich, Chris Miller, and Charles Steinhorn, Structures having o-minimal open core.Trans. Am. Math. Soc., 362(3):1371–1411, (2010)","work_id":"42532b51-913e-4918-9a6d-7d11d98b6dee","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Alfred Dolich, Chris Miller, and Charles Steinhorn, Expansions of o-minimal structures by dense independent sets.Ann. Pure Appl. Logic, 167(8):684–706, (2016)","work_id":"86c87324-a358-470f-bce3-60ad41d4e8b3","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Philipp Hieronymi, Travis Nell, and Erik Walsberg, Wild theories with o-minimal open core.Ann. Pure Appl. Logic, (2) 169:146–163, (2018)","work_id":"294e3dce-bbe7-4243-84b1-58947123a720","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":9,"snapshot_sha256":"e2bb3aa9189ff03f2a03d469e037b6f231b76520a3a8803fa61c63c781c9d521","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"}