{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:22:02Z","timestamp":1760170922481},"reference-count":6,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2015,1,8]],"date-time":"2015-01-08T00:00:00Z","timestamp":1420675200000},"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":[[2015,2]]},"abstract":"<jats:p>We show:\n<jats:list list-type=\"explicit-label\">\n<jats:list-item><jats:p><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0003\" \/> iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete.<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>Every infinite subset <jats:italic>X<\/jats:italic> of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0004\" \/> has a countably infinite subset iff every infinite sequentially closed subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0005\" \/> includes an infinite closed subset.<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>The statement \u201c<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0006.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0006\" \/> is sequential\u201d is equivalent to each one of the following propositions:<\/jats:p><\/jats:list-item>\n<\/jats:list>\n<jats:list list-type=\"explicit-label\">\n<jats:list-item><jats:p>Every sequentially closed subset <jats:italic>A<\/jats:italic> of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0007.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0007\" \/> includes a countable cofinal subset <jats:italic>C<\/jats:italic>,<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>for every sequentially closed subset <jats:italic>A<\/jats:italic> of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0008.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0008\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0009.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0009\" \/>is a meager subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0010.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0010\" \/>,<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>for every sequentially closed subset <jats:italic>A<\/jats:italic> of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0011.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0011\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0012.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0012\" \/>,<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>every sequentially closed subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0013.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0013\" \/> is separable,<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>every sequentially closed subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0014.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0014\" \/> is Cantor complete,<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>every complete subspace of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300008-math-0015.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300008:malq201300008-math-0015\" \/> is Cantor complete.<\/jats:p><\/jats:list-item>\n<\/jats:list>\n<\/jats:p>","DOI":"10.1002\/malq.201300008","type":"journal-article","created":{"date-parts":[[2015,1,8]],"date-time":"2015-01-08T18:46:38Z","timestamp":1420742798000},"page":"24-31","source":"Crossref","is-referenced-by-count":3,"title":["On sequentially closed subsets of the real line in"],"prefix":"10.1002","volume":"61","author":[{"given":"Kyriakos","family":"Keremedis","sequence":"first","affiliation":[{"name":"Department of Mathematics University of the Aegean  Karlovassi 83200 Samos Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2015,1,8]]},"reference":[{"key":"e_1_2_4_2_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310029"},{"key":"e_1_2_4_3_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200710018"},{"key":"e_1_2_4_4_1","volume-title":"The Axiom of Choice, Lecture Notes in Mathematics","author":"Herrlich H.","year":"2006"},{"key":"e_1_2_4_5_1","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200108)47:3<423::AID-MALQ423>3.0.CO;2-0"},{"key":"e_1_2_4_6_1","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/059"},{"key":"e_1_2_4_7_1","first-page":"247","article-title":"On Loeb and weakly Loeb Hausdorff spaces","volume":"53","author":"Keremedis K.","year":"2001","journal-title":"Sci. Math. Jpn."}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201300008","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201300008","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T01:50:17Z","timestamp":1696384217000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201300008"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,8]]},"references-count":6,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2015,2]]}},"alternative-id":["10.1002\/malq.201300008"],"URL":"https:\/\/doi.org\/10.1002\/malq.201300008","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1,8]]}}}