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i read that the circle $S^1$ is the only connected compact 1-manifold but don't we have that the interval $I=[0,1]$ is a connected compact 1-manifold and that is not homeomorphic to $S^1$? May be they mean $S^1$ is the only compact not simply connected 1-manifold?

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    As I noted above, it is perfectly possible that they consider manifolds to be those without boundary. What the term means is just a convention, it is not carved in stone, and very, very often it is useful to make conventions to make one's like easier.2012-08-29

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Here is the classification of $1$-manifolds (connected):

  1. $[0,1]/(0 \equiv 1)$, the unit circle: compact, without boundary.
  2. $[0,1]$: compact with a non-connected boundary.
  3. $(0,1)$: non-compact, without boundary.
  4. $[0,1)$: non-compact, with connected boundary.