{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,9]],"date-time":"2024-01-09T00:04:33Z","timestamp":1704758673681},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2003,1,16]],"date-time":"2003-01-16T00:00:00Z","timestamp":1042675200000},"content-version":"vor","delay-in-days":15,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we are concerned with definably, with or without parameters, (Dedekind) complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o\u2010minimal if and only if all their definable subsets are finite disjoint unions of definably connected (definable) subsets. We pay attention to how simply (in terms of the quantifier complexity and\/or usage of parameters) a definable gap in an expansion is so. Next we prove that over parametrically definably complete expansions of ordered fields, all one\u2010to\u2010one definable (with parameters) continuous functions are monotone and open. Moreover, in both parameter and parameter\u2010free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine\u2010Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets.<\/jats:p>","DOI":"10.1002\/malq.200310005","type":"journal-article","created":{"date-parts":[[2003,1,17]],"date-time":"2003-01-17T16:43:11Z","timestamp":1042821791000},"page":"72-82","source":"Crossref","is-referenced-by-count":1,"title":["Expansions of ordered fields without definable gaps"],"prefix":"10.1002","volume":"49","author":[{"given":"Jafar S.","family":"Eivazloo","sequence":"first","affiliation":[]},{"given":"Mojtaba","family":"Moniri","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2003,1,16]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00054-7"},{"key":"e_1_2_1_3_2","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1215\/ijm\/1256044752","article-title":"Ordered fields satisfying Rolle's theorem","volume":"30","author":"Brown R.","year":"1986","journal-title":"Illinois J. Math."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.2307\/2046522"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01171706"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.2307\/2118545"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)00053-G"},{"key":"e_1_2_1_8_2","unstructured":"J. S.EivazlooandM.Moniri Definably Dedekind complete expansions of ordered fields.Abstrac of a contributed talk at LC '01 Bull. Symbolic Logic8 132\u2013133(2002)."},{"key":"e_1_2_1_9_2","doi-asserted-by":"crossref","unstructured":"R.Goldblatt Lectures on the Hyperreals: An Introduction to Nonst and ard Analysis (Springer\u2010Verlag New York 1998).","DOI":"10.1007\/978-1-4612-0615-6"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.2307\/2694977"},{"key":"e_1_2_1_11_2","doi-asserted-by":"crossref","unstructured":"R.Kaye Models of Peano Arithmetic (Oxford University Press Oxford 1991).","DOI":"10.1093\/oso\/9780198532132.001.0001"},{"key":"e_1_2_1_12_2","doi-asserted-by":"crossref","unstructured":"H.\u2010J.Keisler Monotone complete fields. In: Victoria Symposium on Nonstanard Analysis (A. Hurd and P. Loeb eds.) 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In: Model Theory of Fields (D. Marker M. Messmer and A. Pillay eds.) Lecture Notes in Logic5 pp. 1 \u2013 37 (Springer\u2010Verlag Berlin 1996).","DOI":"10.1007\/978-3-662-22174-7_1"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.2307\/2694974"},{"key":"e_1_2_1_18_2","first-page":"193","article-title":"Expansions of the real line by open sets: o\u2010minimality and open cores","volume":"162","author":"Miller C.","year":"1999","journal-title":"Fund. Math."},{"key":"e_1_2_1_19_2","unstructured":"M.Moniri andJ.S.Eivazloo Using nets in Dedekind monotone or Scott incomplete ordered fields and definability issues. Proceedings of the Ninth Prague Topological Symposium (P. Simon ed.) Topology Atlas (available electronically at http:\/\/at.yorku.ca) Toronto 2002 pp. 195 \u2013 203."},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610799007528"},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833697-X"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.2307\/2695102"},{"key":"e_1_2_1_23_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02567077"},{"key":"e_1_2_1_24_2","unstructured":"D.Scott On completing ordered fields. In: Applications of Model Theory to Algebra Analysis and Probability (W. A. J. 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