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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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

I just edited your question slightly. You might want to click on the link after "edited" to make sure I didn't distort your meaning. :) This question has been asked at least twice before [1] [2]. I don't know if there's a full proof anywhere on this site, but there are a few different references in the comments/answers to those questions. – Micah Feb 4 '13 at 11:19

2 Answers 2

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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Why down-vote ? – Damien L Feb 4 '13 at 12:01
There are 4 connected 1-manifolds: the line, the circle, the long line, and the open long ray. – aaa Feb 4 '13 at 12:10
Are you serious ? The last two ones are not paracompact... So if you want to take it like this, YOU ARE WRONG because there are other non-Haussdorf manifolds. – Damien L Feb 4 '13 at 12:14

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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