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We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space (X,τ) 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.arXiv:1807.08086v2 fatcat:cj6af5pk7zauhplz2y2cslcyju