{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T15:29:56Z","timestamp":1759073396376},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1472,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We work in set-theory without choice <jats:italic><jats:bold>ZF<\/jats:bold><\/jats:italic>. A set is <jats:italic>countable<\/jats:italic> if it is finite or equipotent with \u2115. Given a closed subset <jats:italic>F<\/jats:italic> of [0, 1]<jats:sup>I<\/jats:sup> which is a bounded subset of <jats:italic>\u2113<\/jats:italic><jats:sup>1<\/jats:sup>(<jats:italic>I<\/jats:italic>) (<jats:italic>resp.<\/jats:italic> such that <jats:italic>F<\/jats:italic> \u2286 <jats:italic>c<\/jats:italic><jats:sub>0<\/jats:sub>(<jats:italic>I<\/jats:italic>)), we show that the countable axiom of choice for finite sets, (<jats:italic>resp.<\/jats:italic> the countable axiom of choice <jats:italic><jats:bold>AC<\/jats:bold><\/jats:italic><jats:sub>\u2115<\/jats:sub>) implies that <jats:italic>F<\/jats:italic> is compact. This enhances previous results where <jats:italic><jats:bold>AC<\/jats:bold><\/jats:italic><jats:sub>\u2115<\/jats:sub> (<jats:italic>resp.<\/jats:italic> the axiom of Dependent Choices) was required. If <jats:italic>I<\/jats:italic> is linearly orderable (for example <jats:italic>I<\/jats:italic> = \u211d), then, in <jats:italic><jats:bold>ZF<\/jats:bold><\/jats:italic>, the closed unit ball of the Hilbert space <jats:italic>\u2113<\/jats:italic><jats:sup>2<\/jats:sup> (<jats:italic>I<\/jats:italic>) is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002930_inline1\" \/> is not provable in <jats:italic><jats:bold>ZF<\/jats:bold><\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1264433919","type":"journal-article","created":{"date-parts":[[2010,1,25]],"date-time":"2010-01-25T15:38:59Z","timestamp":1264433939000},"page":"255-268","source":"Crossref","is-referenced-by-count":3,"title":["Notions of compactness for special subsets of \u211d<sup><i>I<\/i><\/sup> and some weak forms of the axiom of choice"],"prefix":"10.1017","volume":"75","author":[{"given":"Marianne","family":"Morillon","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002930_ref014","first-page":"179","article-title":"On continuous images of Eberlein compacts","volume":"17","author":"Simon","year":"1976","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S0022481200002930_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200010)46:4<569::AID-MALQ569>3.0.CO;2-J"},{"key":"S0022481200002930_ref008","doi-asserted-by":"publisher","DOI":"10.4064\/fm-37-1-75-76"},{"key":"S0022481200002930_ref007","volume-title":"The Axiom of Choice","author":"Jech","year":"1973"},{"key":"S0022481200002930_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/059"},{"key":"S0022481200002930_ref004","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610798005675"},{"key":"S0022481200002930_ref003","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19850312507"},{"key":"S0022481200002930_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-06-08527-3"},{"key":"S0022481200002930_ref005","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1521-3870(200001)46:1<3::AID-MALQ3>3.0.CO;2-E"},{"key":"S0022481200002930_ref012","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1965.11970596"},{"key":"S0022481200002930_ref011","first-page":"247","article-title":"On Loeb and weakly Loeb Hausdorff spaces","volume":"53","author":"Keremedis","year":"2001","journal-title":"Scientiae Mathematicae Japonicae"},{"key":"S0022481200002930_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/ba53-4-1"},{"key":"S0022481200002930_ref002","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1977.70.309"},{"key":"S0022481200002930_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2008.01.008"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200002930","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,28]],"date-time":"2019-04-28T20:37:28Z","timestamp":1556483848000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200002930\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,3]]}},"alternative-id":["S0022481200002930"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1264433919","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,3]]}}}