## Topological Methods in Nonlinear Analysis

### Michael's selection theorem for a mapping definable in an o-minimal structure defined on a set of dimesion 1

#### Abstract

Let $R$ be a real closed field and let some o-minimal structure extending $R$ be given. Let $F\colon X \rightrightarrows R^m$ be a definable multivalued lower semicontinuous mapping with nonempty definably connected values defined on a definable subset $X$ of $R^n$ of dimension $1$ ($X$ can be identified with a finite graph immersed in $R^n$). Then $F$ admits a definable continuous selection.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 49, Number 1 (2017), 377-380.

Dates
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1491876035

Digital Object Identifier
doi:10.12775/TMNA.2016.092

Mathematical Reviews number (MathSciNet)
MR3635651

Zentralblatt MATH identifier
1372.14050

#### Citation

Czapla, Małgorzata; Pawłucki, Wiesław. Michael's selection theorem for a mapping definable in an o-minimal structure defined on a set of dimesion 1. Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 377--380. doi:10.12775/TMNA.2016.092. https://projecteuclid.org/euclid.tmna/1491876035

#### References

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• \bibitem 2 M. Czapla and W. Pawłucki, Michael's theorem for Lipschitz cells in o-minimal structures, Ann. Polon. Math. 117 (2016), no. 2, 101–107.
• \bibitem 3 L. van den Dries, Tame Topology and O-minimal Structures, Cambridge University Press, 1998.
• \bibitem 4 E. Michael, Continuous selections \romII, Ann. of Math. (2) 64 (1956), 562–580.