{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T08:56:30Z","timestamp":1773392190392,"version":"3.50.1"},"reference-count":20,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2004,2,3]],"date-time":"2004-02-03T00:00:00Z","timestamp":1075766400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2004,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study within the framework of Zermelo\u2010Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindel\u00f6f metric spaces. We show that in ZF the weak choice principles: (i) Every Lindel\u00f6f metric space is separable and (ii) Every Lindel\u00f6f metric space is second countable (Forms 340 and 341, respectively, in [10]) are equivalent. We also prove that a Lindel\u00f6f metric space is hereditarily separable iff it is hereditarily Lindel\u00f6f iff it hold as well the axiom of choice restricted to countable sets and to topologies of Lindel\u00f6f metric spaces as the countable union theorem restricted to Lindel\u00f6f metric spaces. (\u00a9 2004 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200310084","type":"journal-article","created":{"date-parts":[[2004,2,4]],"date-time":"2004-02-04T13:02:17Z","timestamp":1075899737000},"page":"141-151","source":"Crossref","is-referenced-by-count":9,"title":["Consequences of the failure of the axiom of choice in the theory of Lindel\u00f6f metric spaces"],"prefix":"10.1002","volume":"50","author":[{"given":"Kyriakos","family":"Keremedis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2004,2,3]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19850312507"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-8641(97)00138-7"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.2307\/1989285"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310004"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310049"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-8641(95)90010-1"},{"key":"e_1_2_1_8_2","first-page":"319","article-title":"Products of Lindel\u00f6f T2\u2010spaces are Lindel\u00f6f \u2013 in some models of ZF","volume":"43","author":"Herrlich H.","year":"2002","journal-title":"Comment. Math. Univ. Carolinae"},{"key":"e_1_2_1_9_2","first-page":"553","article-title":"When is \u2115 Lindel\u00f6f","volume":"38","author":"Herrlich H.","year":"1998","journal-title":"Comment. Math. Univ. Carolinae"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200108)47:3<423::AID-MALQ423>3.0.CO;2-0"},{"key":"e_1_2_1_11_2","doi-asserted-by":"crossref","unstructured":"P.Howard andJ. E.Rubin Consequences of the Axiom of Choice. Amer. Math. Soc. Surveys and Monographs Vol. 59 (American Mathematical Society Reading Mass. 1998).","DOI":"10.1090\/surv\/059"},{"key":"e_1_2_1_12_2","unstructured":"T.Jech The axiom of Choice (North\u2010Holland Publ. Comp. Amsterdam 1973)."},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310017"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1521-3870(200001)46:1<35::AID-MALQ35>3.0.CO;2-G"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200101)47:1<117::AID-MALQ117>3.0.CO;2-N"},{"key":"e_1_2_1_16_2","first-page":"247","article-title":"On Loeb and weakly Loeb Hausdorff spaces","volume":"53","author":"Keremedis K.","year":"2001","journal-title":"Scientiae Mathematicae Japonicae"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.2307\/2314411"},{"key":"e_1_2_1_18_2","doi-asserted-by":"crossref","first-page":"49","DOI":"10.4064\/fm-41-1-49-56","article-title":"Effectiveness of the representation theory for Boolean algebras","volume":"41","author":"\u0141o\u015b J.","year":"1954","journal-title":"Fund. Math."},{"key":"e_1_2_1_19_2","unstructured":"H.Rubin andJ. E.Rubin Equivalents of the Axiom of Choice II (North\u2010Holland Publ. Comp. Amsterdam 1985)."},{"key":"e_1_2_1_20_2","first-page":"165","article-title":"Disasters in metric topology without choice","volume":"43","author":"Tachtsis E.","year":"2002","journal-title":"Comment. Math. Univ. Carolinae"},{"key":"e_1_2_1_21_2","unstructured":"S.Willard General Topology (Addison\u2010Wesley Publ. Co. 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