{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T13:33:52Z","timestamp":1765805632939},"reference-count":6,"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":1288,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if <jats:italic>X, Y<\/jats:italic> and <jats:italic>S<\/jats:italic> are non-empty definable sets and <jats:italic>S<\/jats:italic> is a large subset of <jats:italic>X \u00d7 Y.<\/jats:italic> then for a large set of tuples <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002413_inline1\" \/>, where <jats:italic>k<\/jats:italic> = dim(<jats:italic>Y<\/jats:italic>), the union of fibers <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002413_inline2\" \/> is large in <jats:italic>Y.<\/jats:italic> Finally, given a weakly o-minimal structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002413_inline3\" \/>, we find various conditions equivalent to the fact that the topological dimension in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002413_inline3\" \/> enjoys the addition property.<\/jats:p>","DOI":"10.2178\/jsl\/1278682203","type":"journal-article","created":{"date-parts":[[2010,7,9]],"date-time":"2010-07-09T13:30:45Z","timestamp":1278682245000},"page":"841-867","source":"Crossref","is-referenced-by-count":11,"title":["Topological properties of sets definable in weakly o-minimal structures"],"prefix":"10.1017","volume":"75","author":[{"given":"Roman","family":"Wencel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002413_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525919"},{"key":"S0022481200002413_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(88)90125-9"},{"key":"S0022481200002413_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90061-4"},{"key":"S0022481200002413_ref006","first-page":"161","volume-title":"Algebra and model theory, 3 (Erlogol, 2001)","author":"Verbovskiy","year":"2001"},{"key":"S0022481200002413_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-00-02633-7"},{"key":"S0022481200002413_ref001","first-page":"8","volume-title":"Algebra and model theory II","author":"Arefiev","year":"1997"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200002413","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T21:38:38Z","timestamp":1556401118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200002413\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,9]]}},"alternative-id":["S0022481200002413"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1278682203","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,9]]}}}