{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:39:12Z","timestamp":1760236752121,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T00:00:00Z","timestamp":1640217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Sciences, Innovation and Universities of Spain","award":["PGC2018-094431-B-I00"],"award-info":[{"award-number":["PGC2018-094431-B-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Ferrando and L\u00fcdkovsky proved that for a non-empty set \u03a9 and a normed space X, the normed space c0(\u03a9,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n\u2208N defining its topology, then c0(\u03a9,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:\u03a9\u2192X such that for each \u03b5&gt;0 and n\u2208N the set \u03c9\u2208\u03a9:f(\u03c9)n&gt;\u03b5 is finite or empty, with the topology defined by the semi-norms fn=supf(\u03c9)n:\u03c9\u2208\u03a9, n\u2208N. K\u0105kol, L\u00f3pez-Pellicer and Moll-L\u00f3pez also proved that the metrizable space c0(\u03a9,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(\u03a9,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.<\/jats:p>","DOI":"10.3390\/axioms11010006","type":"journal-article","created":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T10:16:18Z","timestamp":1640254578000},"page":"6","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Baire-Type Properties in Metrizable c0(\u03a9, X)"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1655-2320","authenticated-orcid":false,"given":"Salvador","family":"L\u00f3pez-Alfonso","sequence":"first","affiliation":[{"name":"Departamento de Construcciones Arquitect\u00f3nicas, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3918-1713","authenticated-orcid":false,"given":"Manuel","family":"L\u00f3pez-Pellicer","sequence":"additional","affiliation":[{"name":"IUMPA, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3388-5135","authenticated-orcid":false,"given":"Santiago","family":"Moll-L\u00f3pez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1007\/BF01226496","article-title":"Barrelledness conditions on c0(E)","volume":"31","author":"Marquina","year":"1978","journal-title":"Arch. 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