Context:
I'm studying smooth manifolds, and I'm trying to better understand what can be said about the images of smooth maps $\varphi: M \to N$ of constant rank $r$.
I know that from the constant rank theorem, it follows that local images are submanifolds (i.e.: each $p\in M$ admits some neighborhood $U_p$ for which $\varphi(U_p)$ is an (embedded) submanifold of dimension $r$ of $M$.)
But I wonder about when does this hold globally for $\varphi(M)$.
I know that if I additionally suppose both $M$ compact and $\varphi$ injective, then it holds.
If I just supposes $M$ compact, then by looking in Abraham et al.'s Manifolds, Tensor Analysis, and Applications at the last part of Theorem 3.5.18 , it seems to claim that $\varphi(M)$ is in that case an (embedded) submanifold of dimension $r$. Indeed, point $\text{(i)}$ is satisfied (as $\varphi$ is constant rank), and it is a closed map (as $M$ is compact).
But this seems false to me, as one could apply it to the map drawing the symbol "$\alpha$" in $\mathbb{R}^2$ at constant speed, which would contradict the well known fact that this image is not an (embedded) submanifold (since it's not homeomorphic to $\mathbb{R}^1$ at the crossing point).
What am I misunderstanding in this Theorem 3.5.18?