{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T13:01:18Z","timestamp":1791637278727,"version":"4.3.4"},"reference-count":21,"publisher":"Wiley","issue":"4-5","license":[{"start":{"date-parts":[[2007,7,26]],"date-time":"2007-07-26T00:00:00Z","timestamp":1185408000000},"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":[[2007,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Every second\u2010countable regular topological space <jats:italic>X<\/jats:italic> is metrizable. For a given \u201ccomputable\u201d topological space satisfying an axiom of computable regularity M. Schr\u00f6der [10] has constructed a computable metric. In this article we study whether this metric space (<jats:italic>X, d<\/jats:italic>) can be considered computationally as a subspace of some computable metric space [15]. While Schr\u00f6der's construction is \u201cpointless\u201d, i. e., only sets of a countable base but no concrete points are known, for a computable metric space a concrete dense set of computable points is needed. But there may be no computable points in <jats:italic>X<\/jats:italic>. By converging sequences of basis sets instead of Cauchy sequences of points we construct a metric completion (<jats:styled-content>$ \\tilde X $<\/jats:styled-content>, <jats:styled-content>$ \\tilde d $<\/jats:styled-content>) of a space (<jats:italic>X<\/jats:italic>, <jats:italic>d<\/jats:italic>) together with a canonical representation <jats:styled-content>$ \\tilde X $<\/jats:styled-content>, <jats:styled-content>$ \\tilde \\delta $<\/jats:styled-content>. We show that there is a computable embedding of (<jats:italic>X<\/jats:italic>, <jats:italic>d<\/jats:italic>) in (<jats:styled-content>$ \\tilde X $<\/jats:styled-content>, <jats:styled-content>$ \\tilde d $<\/jats:styled-content>) with computable inverse. Finally, we construct a notation of a dense set of points in (<jats:styled-content>$ \\tilde X $<\/jats:styled-content>, <jats:styled-content>$ \\tilde d $<\/jats:styled-content>) with computable mutual distances and prove that the Cauchy representation of the resulting computable metric space is equivalent to <jats:styled-content>$ \\tilde \\delta $<\/jats:styled-content>. Therefore, every computably regular space has a computable homeomorphic embedding in a computable metric space, which topologically is its completion. By the way we prove a computable Urysohn lemma. (\u00a9 2007 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200710009","type":"journal-article","created":{"date-parts":[[2007,7,26]],"date-time":"2007-07-26T08:46:58Z","timestamp":1185439618000},"page":"381-395","source":"Crossref","is-referenced-by-count":14,"title":["Computable metrization"],"prefix":"10.1002","volume":"53","author":[{"given":"Tanja","family":"Grubba","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthias","family":"Schr\u00f6der","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Klaus","family":"Weihrauch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2007,7,26]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(02)00693-X"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00284-9"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.05.001"},{"key":"e_1_2_1_5_2","unstructured":"R.Engelking General Topology. Sigma Series in Pure Mathematics 6 (Heldermann 1989)."},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","unstructured":"T.Grubba andK.Weihrauch A computable version of Dini's theorem for topological spaces. In: Computer and Information Sciences ISCIS 2005 (P. Yolum T. G\u00fcng\u00f6r F. G\u00fcrgen and C. \u00d6zturan eds.). Lecture Notes in Computer Science 3733 pp. 927\u2013936 (Springer 2005).","DOI":"10.1007\/11569596_94"},{"key":"e_1_2_1_7_2","unstructured":"T.Grubba andK.Weihrauch On computable metrization. In: CCA 2006 Third International Conference on Computability and Complexity in Analysis (D. Cenzer R. Dillhage T. Grubba and K. Weihrauch eds.). Informatik Berichte 333 pp. 176\u2013191 (FernUniversit\u00e4t Hagen 2006)."},{"key":"e_1_2_1_8_2","doi-asserted-by":"crossref","first-page":"168","DOI":"10.4064\/fm-42-1-168-202","article-title":"Computable functionals","volume":"42","author":"Grzegorczyk A.","year":"1955","journal-title":"Fundamenta Mathematicae"},{"key":"e_1_2_1_9_2","first-page":"2478","article-title":"Extension de la notion de fonction r\u00e9cursive aux fonctions d'une ou plusieurs variables r\u00e9elles I","volume":"240","author":"Lacombe D.","year":"1955","journal-title":"Comptes Rendus Acad\u00e9mie des Sciences Paris"},{"key":"e_1_2_1_10_2","doi-asserted-by":"crossref","unstructured":"E.Schechter Handbook of Analysis and Its Foundations (Academic Press 1997).","DOI":"10.1016\/B978-012622760-4\/50002-9"},{"key":"e_1_2_1_11_2","unstructured":"M.Schr\u00f6der Effective metrization of regular spaces. In: Computability and Complexity in Analysis (J. Wiedermann K.\u2010I. Ko A. Nerode M. B. Pour\u2010El and K. Weihrauch eds.). Informatik Berichte 235 pp. 63\u201380 (FernUniversit\u00e4t Hagen 1998)."},{"key":"e_1_2_1_12_2","unstructured":"M.Schr\u00f6der Admissible representations for continuous computations. Ph. D. thesis Fachbereich Informatik FernUniversit \u00e4t Hagen 2002."},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00109-8"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00296-5"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(93)90001-A"},{"key":"e_1_2_1_16_2","doi-asserted-by":"crossref","unstructured":"K.Weihrauch Computable Analysis (Springer 2000).","DOI":"10.1007\/978-3-642-56999-9"},{"key":"e_1_2_1_17_2","unstructured":"K.Weihrauch Multi\u2010functions on multi\u2010represented sets are closed under flowchart programming. In: Computability and Complexity in Analysis (T. Grubba P. Hertling H. Tsuiki and K. Weihrauch eds.). Informatik Berichte 326 pp. 267\u2013300 (FernUniversit\u00e4t Hagen 2005)."},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00301-6"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1145\/792538.792542"},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19960420132"},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200210)48:1 <157::AID-MALQ157>3.0.CO;2-4"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310107"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200710009","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.200710009","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T03:16:37Z","timestamp":1693451797000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.200710009"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,7,26]]},"references-count":21,"journal-issue":{"issue":"4-5","published-print":{"date-parts":[[2007,9]]}},"alternative-id":["10.1002\/malq.200710009"],"URL":"https:\/\/doi.org\/10.1002\/malq.200710009","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,7,26]]}}}