Expansions of real closed fields with the Banach fixed point property

Abstract We study a variant of converses of the Banach fixed point theorem and its connection to tameness in expansions of a real closed field. An expansion of a real closed ordered field is said to have the Banach fixed point property when, for every locally closed definable set , if every definable contraction on has a fixed point, then is closed. Let be an expansion of a real closed field. We prove that if has an o‐minimal open core, then it has the Banach fixed point property; and if is definably complete and has the Banach fixed point property, then it has a locally o‐minimal open core.