Definable one-dimensional topologies in O-minimal structures

@article{Peterzil2020DefinableOT,
  title={Definable one-dimensional topologies in O-minimal structures},
  author={Ya'acov Peterzil and A. Carnero Rosel},
  journal={Archive for Mathematical Logic},
  year={2020},
  volume={59},
  pages={103-125}
}
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $$\left( X,\tau \right) $$ X , τ is definably homeomorphic to an affine definable space (namely, a definable subset of $$M^{n}$$ 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. 

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