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An analogue of the Baire category theorem. (English) Zbl 1306.03018

An expansion of an ordered field is definably complete if every nonempty bounded definable subset of the field has a supremum in the field. The author proves a conjecture of A. Fornasiero and T. Servi [Fundam. Math. 209, No. 3, 215–241 (2010; Zbl 1233.03043)] saying that each such expansion is definably Baire, i.e., there is no definable increasing family, with the positive elements as parameters, of nowhere dense subsets of the field covering the whole field. The arguments mainly involve the order and the topology of the ordered field. This means that definably complete structures are a good generalization of o-minimal expansions of fields.

MSC:

03C64 Model theory of ordered structures; o-minimality
54E52 Baire category, Baire spaces

Citations:

Zbl 1233.03043
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References:

[1] Expansions of dense linear orders with the intermediate value property 66 pp 1783– (2001) · Zbl 1001.03039
[2] DOI: 10.1090/S0002-9939-10-10268-8 · Zbl 1233.03044
[3] Illinois Journal of Mathematics
[4] Transactions of the American Mathematical Society 362 pp 1371– (2010)
[5] DOI: 10.4064/fm209-3-2 · Zbl 1233.03043
[6] DOI: 10.1007/s00153-011-0235-x · Zbl 1220.03029
[7] DOI: 10.1007/978-1-4614-4042-0_6 · Zbl 1253.03064
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