{"title":"Irreducibility of strong size levels","authors":"Alejandro Illanes, Verónica Martínez-de-la-Vega","doi":"10.1515/ms-2024-0039","DOIUrl":null,"url":null,"abstract":"Given a continuum <jats:italic>X</jats:italic>, <jats:italic>C<jats:sub>n</jats:sub> </jats:italic>(<jats:italic>X</jats:italic>) denotes the hyperspace of nonempty closed subsets of <jats:italic>X</jats:italic> with at most <jats:italic>n</jats:italic> components. A strong size level is a subset of the form <jats:italic>σ</jats:italic> <jats:sup>−1</jats:sup>(<jats:italic>t</jats:italic>), where <jats:italic>σ</jats:italic> is a strong size map for <jats:italic>C<jats:sub>n</jats:sub> </jats:italic>(<jats:italic>X</jats:italic>) and <jats:italic>t</jats:italic> ∈ (0, 1]. In this paper, answering a question by Capulín-Pérez, Fuentes-Montes de Oca, Lara-Mejía and Orozco-Zitli, we prove that for each <jats:italic>n</jats:italic> ≥ 2, no strong size level for <jats:italic>C<jats:sub>n</jats:sub> </jats:italic>(<jats:italic>X</jats:italic>) is irreducible.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/ms-2024-0039","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
Given a continuum X, Cn(X) denotes the hyperspace of nonempty closed subsets of X with at most n components. A strong size level is a subset of the form σ−1(t), where σ is a strong size map for Cn(X) and t ∈ (0, 1]. In this paper, answering a question by Capulín-Pérez, Fuentes-Montes de Oca, Lara-Mejía and Orozco-Zitli, we prove that for each n ≥ 2, no strong size level for Cn(X) is irreducible.
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