Definable quotients in locally o-minimal structures

Abstract

Let $\mathcal F=(F, +. \cdot, <, 0, 1, \dots)$ be a definably complete locally o-minimal expansion of an ordered field. We demonstrate the existence of definable quotients of definable sets by definable equivalence relations when several technical conditions are satisfied. These conditions are satisfied when $X$ is a locally closed definable subset of $F^n$ and there is a definable proper action of a definable group $G$ on $X$.