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My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary.

enter image description here

I have been trying to wrap my head around this for about 2 hours (3.5 hours, if you include the 1.5 hours spent on the other question).

An alternative way I approach the example: classification of smooth 1-manifolds with boundary

  1. Does the classification of smooth 1-manifolds with boundary imply $C = c[a,b]$ and $[a,b]$ are diffeomorphic to $[0,1]$ and thus diffeomorphic to each other and thus $\partial C$ is diffeomorphic to $\partial [a,b] = \{a,b\}$, and $C^o$ is diffeomorphic to $(a,b)$?

  2. Must $c$ be an embedding and in particular $c$ is injective?

Update: The classification theorem assumes nonempty boundary. Let's do so as well otherwise the example is wrong. Also see here for the same question but not specifically to do with the classification theorem.

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You are right that the classification does not imply that $C$ and $[a, b]$ are diffeomorphic. It may very well be that $C$ is diffeomorphic to a circle, and that $c$ winds the interval $[a, b]$ 3.5 times around it. (The minute hand of the clock was giving you a hint.)

But assuming that $C$ has nonempty boundary, it follows indeed that the immersion $c$ is injective (i.e. is an embedding), as shown in a linked question: Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?

In particular, the boundary of $C$ has cardinality $2$ and equals $\{ c(a), c(b) \}$.

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