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 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).
