The following exercise corresponds to 2. of Chapter 1, Section 3 of Hirch's book Differential topology.
Let $M$ be a connected non-compact Hausdorff manifold (without boundary) of class $C^r$. Then, there exists a closed $C^r$-embedding of $[0,\infty)$ into $M$.
I have been thinking of this exercise some days and, the more I think about it, the weirder I find it. I have decided to consider $M=\mathbb R^n$. Then the inclusion $[0,1]\hookrightarrow\mathbb R^m$ as $t\mapsto (t,0,\dots,0)$ is clearly a topological embedding. However, the image is not a topological submanifold (without boundary). Then, how could it be a $C^r$-embedding?
In fact, if such a embedding exists, $[0,\infty)$ would be a manifold (without boundary), according to Theorem 3.1.
Am I misunderstanding something or is the exercise wrong?
Thank you.
$M$ may be a manifold without boundary, but $[0,\infty)$ is not, and neither it nor its image are expected to be. That is, the $C^r$-embedding of $[0,\infty)$ into $M$ is intended to be a $C^r$-embedding for manifolds with boundary, of which manifolds without boundary are a sub-category .