{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T15:33:11Z","timestamp":1759073591597},"reference-count":8,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2008,3,7]],"date-time":"2008-03-07T00:00:00Z","timestamp":1204848000000},"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":[[2008,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Under the axiom of choice, every first countable space is a Fr\u00e9chet\u2010Urysohn space. Although, in its absence even \u211d may fail to be a sequential space.<\/jats:p><jats:p>Our goal in this paper is to discuss under which set\u2010theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of \u211d, are classes of Fr\u00e9chet\u2010Urysohn or sequential spaces.<\/jats:p><jats:p>In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion.<\/jats:p><jats:p>Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in \u211d if and only if the axiom of countable choice holds for families of subsets of \u211d, and every metric space has a unique <jats:styled-content>$ \\hat \\sigma $<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-1.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content>\u2010completion. (\u00a9 2008 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200710018","type":"journal-article","created":{"date-parts":[[2008,3,7]],"date-time":"2008-03-07T08:44:58Z","timestamp":1204879498000},"page":"145-152","source":"Crossref","is-referenced-by-count":3,"title":["On countable choice and sequential spaces"],"prefix":"10.1002","volume":"54","author":[{"given":"Gon\u00e7alo","family":"Gutierres","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2008,3,7]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"R.Engelking General Topology revised and completed edition (Heldermann Verlag Berlin 1987)."},{"key":"e_1_2_1_3_2","unstructured":"W.Felscher Naive Mengen und abstrakte Zahlen III \u2013 Transfinite Methoden (Bibliographisches Institut 1979)."},{"key":"e_1_2_1_4_2","unstructured":"A. A.Fraenkel Y.Bar\u2010Hillel andA.Levy Foundations of Set Theory second revised edition (North\u2010Holland 1973)."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760939"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310029"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1080\/16073606.1997.9632237"},{"key":"e_1_2_1_8_2","first-page":"553","article-title":"When \u2115 is Lindel\u00f6f","volume":"38","author":"Herrlich H.","year":"1997","journal-title":"Comment. Math. Univ. Carolinae"},{"key":"e_1_2_1_9_2","doi-asserted-by":"crossref","unstructured":"P.Howard andJ. E.Rubin Consequences of the Axiom of Choice. AMS Math. Surveys and Monographs 59 (1998).","DOI":"10.1090\/surv\/059"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200710018","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.200710018","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T11:35:19Z","timestamp":1697628919000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.200710018"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3,7]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,4]]}},"alternative-id":["10.1002\/malq.200710018"],"URL":"https:\/\/doi.org\/10.1002\/malq.200710018","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3,7]]}}}