Definable continuous selections of set-valued maps in o-minimal expansions of the real field
Volume 65 / 2017
Bulletin Polish Acad. Sci. Math. 65 (2017), 97-105
MSC: Primary 03C64; Secondary 03C98, 26B05, 54C65.
DOI: 10.4064/ba8130-10-2017
Published online: 8 November 2017
Abstract
Let $T$ be a set-valued map from a subset of $\mathbb {R}^n$ to $\mathbb {R}^m$. Suppose $(\mathbb {R};+,\cdot ,T)$ is o-minimal. We prove that (1) if for every $x\in \mathbb {R}^n$, each connected component of $T(x)$ is convex, then $T$ has a continuous selection if and only if $T$ has a continuous selection definable in $(\mathbb {R};+,\cdot ,T)$; (2) if $n=1$ or $m=1$, then $T$ has a continuous selection if and only if $T$ has a continuous selection definable in $(\mathbb {R};+,\cdot ,T)$.
