Michael’s selection theorem in d-minimal expansions of the real field. (English) Zbl 1499.54096
Summary: 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 \( f : E \rightarrow \mathbb{R}^m\) 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.
MSC:
| 54C65 | Selections in general topology |
| 26B05 | Continuity and differentiation questions |
| 03C64 | Model theory of ordered structures; o-minimality |
References:
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