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The only 1-manifolds are $\mathbb R$ and $S^1$

Any manifold is homeomorphic to the disjoint sum of its connected components. Therefore, the full classification of manifolds of dimension 1 reduces to the study of connected manifolds.

Could you please give a proof (sketch) as well or link to a good reference on the subject?

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marked as duplicate by Micah, Davide Giraudo, Chris Eagle, Michael Albanese, Hagen von Eitzen Feb 4 '13 at 13:08

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Any connected manifold of dimension 1 is diffeomorphic to the circle or to the line. If you can read french : Introduction aux variétés différentielles, Jacques Lafontaine.

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Line in the case of smooth manifolds, up to homeomorphism, there are only 2 connected 1-dimensional manifolds: the circle and the real line. You can find the proof of that fact - at least in the smooth case - in Milnor's classical text "Topology from a differentiable viewpoint", which is something always worth reading, and also in many places in the internet, for example here.

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