What is the topological classification of connected 1-manifolds?

Question

Possible Duplicate:
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?

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 04 '13 at 11:19
another question with sketch of proof https://math.stackexchange.com/questions/1866339/how-to-prove-that-there-are-only-two-kinds-of-1-dim-manifolds-without-boundary — Nikolay, Jun 07 '21 at 12:35

score 1 · Answer 1 · answered Feb 04 '13 at 11:11

1

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.

answered Feb 04 '13 at 11:11

Damien L

6,649

2

Why down-vote ? – Damien L Feb 04 '13 at 12:01
There are 4 connected 1-manifolds: the line, the circle, the long line, and the open long ray. – aaa Feb 04 '13 at 12:10
6

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 04 '13 at 12:14

score 1 · Answer 2 · answered Feb 04 '13 at 11:13

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.

What is the topological classification of connected 1-manifolds?

2 Answers2

Linked