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

Thamrongthanyalak, Athipat

doi:10.1090/proc/14283

Michael’s Selection Theorem in d-minimal expansions of the real field
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Proc. Amer. Math. Soc. 147 (2019), 1059-1071 Request permission

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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