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?
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.