{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:40:21Z","timestamp":1777516821516,"version":"3.51.4"},"reference-count":10,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2019,9,27]],"date-time":"2019-09-27T00:00:00Z","timestamp":1569542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2020,2,26]]},"abstract":"<jats:p>We investigate the statement \u201cthe order topology of every countable complete linear order is compact\u201d in the framework of reverse mathematics, and we find that the statement\u2019s strength depends on the precise formulation of compactness. If we require that open covers must be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to [Formula: see text] over [Formula: see text]. If open covers need not be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to [Formula: see text] over [Formula: see text]. This answers a question of Fran\u00e7ois Dorais.<\/jats:p>","DOI":"10.3233\/com-190262","type":"journal-article","created":{"date-parts":[[2019,9,27]],"date-time":"2019-09-27T14:10:55Z","timestamp":1569593455000},"page":"25-36","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":3,"title":["The strength of compactness for countable complete linear orders"],"prefix":"10.1177","volume":"9","author":[{"given":"Paul","family":"Shafer","sequence":"first","affiliation":[{"name":"School of Mathematics, University of Leeds, Leeds, United Kingdom."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2019,9,27]]},"reference":[{"key":"ref001","unstructured":"F.G.\u00a0Dorais, Reverse mathematics of compact countable second-countable spaces, 2011, arXiv:1110.6555v1."},{"key":"ref002","unstructured":"F.G.\u00a0Dorais, 2018, Personal communication."},{"key":"ref003","unstructured":"H.\u00a0Friedman, Some systems of second order arithmetic and their use, in: Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que, 1975, pp.\u00a0235\u2013242."},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-015-0473-4"},{"key":"ref005","unstructured":"P.\u00a0H\u00e1jek and P.\u00a0Pudl\u00e1k, Metamathematics of First-Order Arithmetic, 2nd edn, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998, p.\u00a0xiv+460. ISBN 3-540-63648-X."},{"key":"ref006","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061306000578"},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1130335208"},{"key":"ref008","unstructured":"J.R.\u00a0Munkres, Topology, 2nd edn, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, p.\u00a0xvi+537. ISBN 0-13-181629-2."},{"key":"ref009","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2012.10.014"}],"container-title":["Computability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-190262","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/COM-190262","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-190262","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T16:00:11Z","timestamp":1777392011000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/COM-190262"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,27]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,2,26]]}},"alternative-id":["10.3233\/COM-190262"],"URL":"https:\/\/doi.org\/10.3233\/com-190262","relation":{},"ISSN":["2211-3568","2211-3576"],"issn-type":[{"value":"2211-3568","type":"print"},{"value":"2211-3576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,9,27]]}}}