{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:18:10Z","timestamp":1773249490079,"version":"3.50.1"},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4940,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2000,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>X<\/jats:italic> be a compact metric space. A closed set <jats:italic>K \u2286 X<\/jats:italic> is <jats:italic>located<\/jats:italic> if the distance function <jats:italic>d(x, K)<\/jats:italic> exists as a continuous real-valued function on <jats:italic>X; weakly located<\/jats:italic> if the predicate <jats:italic>d(x, K) &gt; r<\/jats:italic> is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200011993_inline1\"\/> allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA<jats:sub>0<\/jats:sub>, WKL<jats:sub>0<\/jats:sub> and ACA<jats:sub>0<\/jats:sub>. We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA<jats:sub>0<\/jats:sub> version of this result for weakly located closed sets.<\/jats:p>","DOI":"10.2307\/2586708","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:02:41Z","timestamp":1146924161000},"page":"1451-1480","source":"Crossref","is-referenced-by-count":24,"title":["Located sets and reverse mathematics"],"prefix":"10.1017","volume":"65","author":[{"given":"Mariagnese","family":"Giusto","sequence":"first","affiliation":[]},{"given":"Stephen G.","family":"Simpson","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200011993_ref011","first-page":"482","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200011993_ref010","first-page":"402","volume-title":"Classical descriptive set theory","author":"Kechris","year":"1985"},{"key":"S0022481200011993_ref008","first-page":"95\u201396","volume":"39","author":"Jockusch","year":"1974","journal-title":"classes and Boolean combinations of recursively enumerable sets"},{"key":"S0022481200011993_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050103"},{"key":"S0022481200011993_ref005","first-page":"529","volume-title":"General topology","author":"Engelking","year":"1989"},{"key":"S0022481200011993_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90066-7"},{"key":"S0022481200011993_ref012","first-page":"783\u2013802","volume":"49","author":"Simpson","year":"1984","journal-title":"Which set existence axioms are needed to prove the Cauchy\/Peano theorem for ordinary differential equations?"},{"key":"S0022481200011993_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61667-9"},{"key":"S0022481200011993_ref009","first-page":"702","volume-title":"Logic, methodology and philosophy of science VIII","author":"Jockusch","year":"1989"},{"key":"S0022481200011993_ref003","first-page":"297","volume-title":"Logic and computation","author":"Brown","year":"1990"},{"key":"S0022481200011993_ref014","unstructured":"Simpson Stephen G. , Finite and countable additivity, preprint, 8 pages, to appear, November 1996."},{"key":"S0022481200011993_ref015","first-page":"445","volume-title":"Subsystems of second order arithmetic","author":"Simpson","year":"1998"},{"key":"S0022481200011993_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF01621469"},{"key":"S0022481200011993_ref002","unstructured":"Brown Douglas K. , Functional analysis in weak subsystems of second order arithmetic, Ph.D. thesis , The Pennsylvania State University, 1987, vii + 150 pages."},{"key":"S0022481200011993_ref013","first-page":"490","volume-title":"Proof theory","author":"Simpson","year":"1987"},{"key":"S0022481200011993_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200011993_ref006","unstructured":"Giusto Mariagnese , Topology and analysis in reverse mathematics, Ph.D. thesis , Universit\u00e0 di Torino, 1998."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200011993","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T16:46:56Z","timestamp":1557334016000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200011993\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,9]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2000,9]]}},"alternative-id":["S0022481200011993"],"URL":"https:\/\/doi.org\/10.2307\/2586708","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,9]]}}}