Definable one-dimensional topologies in O-minimal structures

Abstract

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $$\left( X,\tau \right) $$ is definably homeomorphic to an affine definable space (namely, a definable subset of $$M^{n}$$ with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.