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Michael’s Selection Theorem in d-minimal expansions of the real field


Author: Athipat Thamrongthanyalak
Journal: Proc. Amer. Math. Soc. 147 (2019), 1059-1071
MSC (2010): Primary 26B05; Secondary 03C64
DOI: https://doi.org/10.1090/proc/14283
Published electronically: December 3, 2018
MathSciNet review: 3896056
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Abstract: Let $E \subseteq \mathbb {R}^n$. If $T$ is a lower semi-continuous set-valued map from $E$ to $\mathbb {R}^m$ and $(\mathbb {R},+,\cdot ,T)$ is d-minimal, then there is a continuous function $f \colon E \to \mathbb {R}^m$ definable in $(\mathbb {R},+,\cdot ,T)$ such that $f(x) \in T(x)$ for every $x \in E$. To prove this result, we establish a cell decomposition theorem for d-minimal expansions of the real field.


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Athipat Thamrongthanyalak
Affiliation: Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10400, Thailand
Email: athipat.th@chula.ac.th

Received by editor(s): March 8, 2017
Received by editor(s) in revised form: March 5, 2018
Published electronically: December 3, 2018
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2018 American Mathematical Society