Expansions of real closed fields with the Banach fixed point property

Pub Date : 2024-07-15 DOI:10.1002/malq.202400001

Athipat Thamrongthanyalak

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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 $E$ , if every definable contraction on $E$ has a fixed point, then $E$ is closed. Let $R$ be an expansion of a real closed field. We prove that if $R$ has an o-minimal open core, then it has the Banach fixed point property; and if $R$ is definably complete and has the Banach fixed point property, then it has a locally o-minimal open core.

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具有巴拿赫定点特性的实闭域展开

我们研究巴拿赫定点定理会话的一个变体及其与实闭域展开中的驯服性的联系。对于每个局部封闭的可定义集合 , 如果其上的每个可定义收缩都有一个定点，则称实闭有序域的展开具有巴拿赫定点性质。设是一个实封闭域的展开式。我们证明，如果有一个 o-minimal 开核，那么它具有巴拿赫定点性质；如果是可定义完全且具有巴拿赫定点性质，那么它有一个局部 o-minimal 开核。

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