{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T15:23:50Z","timestamp":1759073030873},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1288,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We establish the following results:<\/jats:p><jats:p>1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set <jats:italic>I<\/jats:italic> and for every ordinal number \u03b1 \u2265 \u03c9, the following statements are equivalent:<\/jats:p><jats:p>(a) <jats:italic>The Tychonoff product of \u2223\u03b1\u2223 many non-empty finite discrete subsets of I is compact<\/jats:italic>.<\/jats:p><jats:p>(b) <jats:italic>The union of \u2223\u03b1\u2223 many non-empty finite subsets of I is well orderable<\/jats:italic>.<\/jats:p><jats:p>2. The statement: <jats:italic>For every infinite set I, every closed subset of the Tychonoff product<\/jats:italic> [0, 1]<jats:sup>I<\/jats:sup><jats:italic>which consists offunctions with finite support is compact<\/jats:italic>, is not provable in ZF set theory.<\/jats:p><jats:p>3. The statement: <jats:italic>For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact<\/jats:italic>, is not provable in ZF<jats:sup>0<\/jats:sup> (i.e., ZF minus the Axiom of Regularity).<\/jats:p><jats:p>4. The statement: <jats:italic>For every set I, every<\/jats:italic> \u2135<jats:sub>0<\/jats:sub>-<jats:italic>sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of<\/jats:italic> \u2135<jats:sub>0<\/jats:sub><jats:italic>many non-empty finite discrete subsets of I is compact<\/jats:italic>, is not provable in ZF<jats:sup>0<\/jats:sup>.<\/jats:p>","DOI":"10.2178\/jsl\/1278682212","type":"journal-article","created":{"date-parts":[[2010,7,9]],"date-time":"2010-07-09T09:30:45Z","timestamp":1278667845000},"page":"996-1006","source":"Crossref","is-referenced-by-count":4,"title":["Products of some special compact spaces and restricted forms of AC"],"prefix":"10.1017","volume":"75","author":[{"given":"Kyriakos","family":"Keremedis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eleftherios","family":"Tachtsis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002504_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02007144"},{"key":"S0022481200002504_ref014","unstructured":"Morillon M. , Notions of compactness for special subsets of \u211dI and some weak forms of the Axiom of Choice, communicated manuscript."},{"key":"S0022481200002504_ref012","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1965.11970596"},{"key":"S0022481200002504_ref011","doi-asserted-by":"publisher","DOI":"10.4064\/ba53-4-1"},{"key":"S0022481200002504_ref003","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310004"},{"key":"S0022481200002504_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-37-1-75-76"},{"key":"S0022481200002504_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2008.01.008"},{"key":"S0022481200002504_ref007","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1521-3870(200001)46:1<3::AID-MALQ3>3.0.CO;2-E"},{"key":"S0022481200002504_ref004","first-page":"143","volume":"67","author":"De la Cruz","year":"2002","journal-title":"Definitions of compactness and the axiom of choice"},{"key":"S0022481200002504_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0166-8641(95)90010-1"},{"key":"S0022481200002504_ref006","first-page":"645","volume":"55","author":"Howard","year":"1990","journal-title":"Definitions of compact"},{"key":"S0022481200002504_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/059"},{"key":"S0022481200002504_ref009","volume-title":"The Axiom of Choice","author":"Jech","year":"1973"},{"key":"S0022481200002504_ref015","unstructured":"Morillon M. , Synth\u00e8se, preprint, (see http:\/\/personnel.univ-reunion.fr\/mar)."},{"key":"S0022481200002504_ref016","first-page":"439","article-title":"Two remarks on Tychonoff's product theorem","volume":"12","author":"Mycielski","year":"1964","journal-title":"Bulletin de l'Academie Polonaise des Sciences, S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S0022481200002504_ref002","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200211)48:4<508::AID-MALQ508>3.0.CO;2-#"},{"key":"S0022481200002504_ref017","first-page":"389","article-title":"Some topological theorems equivalent to the Boolean prime ideal theorem","volume":"60","author":"Rubin","year":"1954","journal-title":"Bulletin of the American Mathematical Society"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200002504","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T17:39:13Z","timestamp":1556386753000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200002504\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,9]]}},"alternative-id":["S0022481200002504"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1278682212","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,9]]}}}