{"paper":{"title":"The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.GN","authors_text":"Carmelo A. Finocchiaro, Dario Spirito, Marco Fontana","submitted_at":"2018-05-31T13:16:31Z","abstract_excerpt":"Given an arbitrary spectral space $X$, we consider the set ${\\boldsymbol{\\mathcal{X}}}(X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on ${\\boldsymbol{\\mathcal{X}}}(X)$ and, after observing that it coincides the upper Vietoris topology, we prove that ${\\boldsymbol{\\mathcal{X}}}(X)$ is itself a spectral space, that this construction is functorial, and that ${\\boldsymbol{\\mathcal{X}}}(X)$ provides an extension of $X$ in a more `complete' spectral space. Among the applications, we show that, starting from an integral d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.12454","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"}