Submission date: 26 April 2020

Abstract:

An expansion of a real closed ordered field is said to have the Banach Fixed Point Property when for every locally closed definable set E, if every definable contraction on E has a fixed point, then E is closed. We prove that an expansion of a real closed ordered field has o-minimal open core if and only if it is definably complete and has the Banach Fixed Point Property. As consequences, we obtain that the possession of o-minimal open cores is a first-order property in languages extending the language of the ordered rings and is preserved under elementary equivalence.

Mathematics Subject Classification: Primary 03C64 Secondary 54H25, 47H10

Keywords and phrases: Banach Fixed Point Theorem, o-minimal open cores, definable completeness

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