  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.
