{"paper":{"title":"Disconnectedness properties of Hyperspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Angel Tamariz-Mascar\\'ua, Rodrigo Hern\\'andez-Guti\\'errez","submitted_at":"2018-09-18T16:00:10Z","abstract_excerpt":"Let $X$ be a Hausdorff space and let $\\mathcal{H}$ be one of the hyperspaces $CL(X)$, $\\mathcal{K}(X)$, $\\mathcal{F}(X)$ or $\\mathcal{F}_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\\mathcal{H}$: extremal disconnectedness, being a $F^\\prime$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\\subset X$ is a closed subset such that $(a)$ both $F$ and $X-F$ are totally disconnected, $(b)$ the quotient $X/F$ is hereditarily disconnected, then $\\mathcal{K}(X)$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.06807","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"}