Should I say anything else to prove the following from Professor Lee's Intro to Smooth Manifolds text? Thank you.
Prove Proposition 5.21 (sufficient conditions for immersed submanifolds to be embedded). Suppose $M$ is a smooth manifold with or without boundary, and $S\subseteq M$ is an immersed submanifold. If any of the following holds, then $S$ is embedded.
(a) $S$ has codimension $0$ in $M.$
(b) The inclusion map $S\subseteq M$ is proper.
(c) S is compact.
$\textit{Proof.}$ If $(a)$ holds then Theorem 4.5 (Inverse Function Theorem for Manifolds) shows that the inclusion map $\iota : S\hookrightarrow M$ is an open map. The result follows from Proposition 4.22 where we suppose $M$ and $N$ are smooth manifolds with or without boundary, and $F: M\to N$ is an injective smooth immersion. If any of the following holds, then $F$ is a smooth embedding.
(a) $F$ is an open or closed map.
(b) $F$ is a proper map.
(c) $M$ is compact.
(d) $M$ has empty boundary and $\dim M = \dim N.$
The statement is false, at least (c). Take $f : \mathbb{S}^1 \to \mathbb{R}^2$ defined by $f(\cos \theta, \sin \theta) = (\cos \theta, \sin 3\theta)$. This is an immersion, $\mathbb{S}^1$ is compact, but $f$ is not an embedding.