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

DOI: http://dx.doi.org/10.12775/TMNA.2016.092

#### 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.

#### Keywords

Michael's selection theorem; o-minimal structure; finite graph

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