{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,24]],"date-time":"2023-09-24T04:25:14Z","timestamp":1695529514951},"reference-count":9,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2018,4,16]],"date-time":"2018-04-16T00:00:00Z","timestamp":1523836800000},"content-version":"vor","delay-in-days":15,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2018,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the realm of Lindel\u00f6f metric spaces the following results are obtained in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0001\" \/>: (i) If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0002\" \/> is a Lindel\u00f6f metric space then it is both densely Lindel\u00f6f and almost Lindel\u00f6f. In addition, under the countable axiom of choice <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0003\" \/>, the three notions coincide. (ii) The statement \u201cevery separable metric space is almost Lindel\u00f6f\u201d implies that every infinite subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0004\" \/> has a countably infinite subset). (iii) The statement \u201cevery almost Lindel\u00f6f metric space <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0005\" \/> is <jats:italic>quasi totally bounded<\/jats:italic> implies <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0006.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0006\" \/>. (iv) The proposition \u201cevery quasi totally bounded metric space is separable\u201d lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0007.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0007\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0008.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0008\" \/>. Likewise, the statement \u201cevery pre\u2010Lindel\u00f6f (or Lindel\u00f6f) metric space is separable\u201d lies strictly between <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0009.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0009\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600059-math-0010.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600059:malq201600059-math-0010\" \/>.<\/jats:p>","DOI":"10.1002\/malq.201600059","type":"journal-article","created":{"date-parts":[[2018,4,17]],"date-time":"2018-04-17T04:22:36Z","timestamp":1523938956000},"page":"37-43","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two new equivalents of Lindel\u00f6f metric spaces"],"prefix":"10.1002","volume":"64","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":[[2018,4,16]]},"reference":[{"key":"e_1_2_5_2_1","first-page":"161","article-title":"Lindel\u00f6f R\u00e4ume und Auswahlaxiom","volume":"119","author":"Brunner N.","year":"1982","journal-title":"Anz. \u00d6sterr. 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