{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T14:41:08Z","timestamp":1649083268686},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11699,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,3]]},"abstract":"<jats:p>The recursion-theoretic study of mathematical structures other than<jats:italic>\u03c9<\/jats:italic>is now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombe's inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabin's study of computable fields [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]\u2013[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others.<\/jats:p><jats:p>Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological space<jats:italic>X<\/jats:italic>with a countable basis \u22bf for the topology. The space is<jats:italic>fully effective<\/jats:italic>, that is, the basis elements are coded into<jats:italic>\u03c9<\/jats:italic>and the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044790_inline01\" \/>, the lattice of recursively enumerable (r.e.) subsets of<jats:italic>\u03c9<\/jats:italic>, the collection of r.e. open subsets of<jats:italic>X<\/jats:italic>forms a lattice<jats:italic>\u2112(X)<\/jats:italic>under the usual operations of union and intersection.<\/jats:p>","DOI":"10.2307\/2273390","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:58:32Z","timestamp":1146952712000},"page":"169-183","source":"Crossref","is-referenced-by-count":11,"title":["Simplicity in effective topology"],"prefix":"10.1017","volume":"47","author":[{"given":"Iraj","family":"Kalantari","sequence":"first","affiliation":[]},{"given":"Anne","family":"Leggett","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044790_ref014","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1944-08111-1"},{"key":"S0022481200044790_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90015-8"},{"key":"S0022481200044790_ref008","first-page":"1040","article-title":"Les ensembles r\u00e9cursivement ouverts ou ferm\u00e9s, et leurs applications \u00e0 l'analyse r\u00e9cursive","volume":"245","author":"Lacombe","year":"1957","journal-title":"Comptes Rendus Hebdomadaires des S\u00e9ances de l'Acad\u00e9mie des Sciences (Paris)"},{"key":"S0022481200044790_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0227009-1"},{"key":"S0022481200044790_ref001","first-page":"407","article-title":"Effective procedures in field theory","volume":"284","author":"Fr\u00f6lich","year":"1955","journal-title":"Philosophical Transactions of the Royal Society of London, Series A"},{"key":"S0022481200044790_ref002","volume-title":"Proceedings of the European Summer Meeting of the Association for Symbolic Logic, Patras, Greece","author":"Kalantari"},{"key":"S0022481200044790_ref004","unstructured":"Kalantari I. and Remmel J. , Degrees of unsolvability in effective topology (to appear)."},{"key":"S0022481200044790_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90011-1"},{"key":"S0022481200044790_ref007","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-68-03513-8"},{"key":"S0022481200044790_ref016","first-page":"145","volume":"14","author":"Specker","year":"1949","journal-title":"Nicht konstraktiv beweisbare S\u00e4tze der Analysis"},{"key":"S0022481200044790_ref009","first-page":"405","volume":"41","author":"Lerman","year":"1976","journal-title":"Congruence relations, filters, ideals and definability in lattice of \u03b1-recursively enumerable sets"},{"key":"S0022481200044790_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1980-0549153-7"},{"key":"S0022481200044790_ref011","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BFb0062858","volume-title":"Algebra and Logic, Lecture Notes in Mathematics","volume":"450","author":"Metakides","year":"1975"},{"key":"S0022481200044790_ref003","volume-title":"Proceedings of the European Summer Meeting of the Association for Symbolic Logic, Patras, Greece","author":"Kalantari"},{"key":"S0022481200044790_ref005","first-page":"609","volume":"44","author":"Kalantari","year":"1979","journal-title":"Recursive constructions in topological spaces"},{"key":"S0022481200044790_ref015","first-page":"341","article-title":"Computable algebra, general theory and theory of computable fields","volume":"95","author":"Rabin","year":"1960","journal-title":"Transactions of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044790","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T14:18:07Z","timestamp":1586873887000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044790\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,3]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1982,3]]}},"alternative-id":["S0022481200044790"],"URL":"https:\/\/doi.org\/10.2307\/2273390","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}