Definable continuous selections of set-valued maps in o-minimal expansions of the real field

Saronsad Sokantika ¹   ; Athipat Thamrongthanyalak ¹

¹ Department of Mathematics and Computer Science Faculty of Science Chulalongkorn University Bangkok 10330, Thailand

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 65 (2017) no. 2, pp. 97-105

Résumé

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 $(\mathbb {R};+,\cdot ,T)$.

Détail

DOI : 10.4064/ba8130-10-2017

Keywords: set valued map subset mathbb mathbb suppose mathbb cdot o minimal prove nbsp every mathbb each connected component convex has continuous selection only has continuous selection definable mathbb cdot nbsp has continuous selection only has continuous selection definable mathbb cdot