Definable proper quotients (Model theoretic aspects of the notion of independence and dimension)

Abstract

Consider a definably complete locally o-minimal expansion F = (F, +.·, <, 0, 1, ...) of an ordered field. We prove the existence of definable quotients of definable sets by definable equivalence relations when curtain 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. We give its application.