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I want to find a proof of $S^1$ is the only compact connected 1-manifold. (In here, manifold means Hausdorff, locally euclidean space) Is there any reference? Or is it easy and can be proved simply?

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A proof can be found in the appendix of Milnor's Topology from the Differentiable Viewpoint. – Henry T. Horton Oct 26 '12 at 3:45
Milnor only covers the case of smooth manifolds, but the result is the same for topological. You can find an unpublished write-up of a proof by googling classification of 1-manifolds. – Kevin Carlson Oct 26 '12 at 3:48
up vote 2 down vote accepted

You can find a proof in Introduction to Topological Manifolds by John Lee. It is a section on the classification of $1$-dimensional manifolds under the chapter on cell complexes.

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