It is well known that if $X$ and $Y$ are closed, connected $n$-manifolds, then any embedding $f:X \to Y$ is surjective and so a homeomorphism. The standard way to see this is to notice that $f(X)$ must be both open (invariance of domain) and closed ($X$ is compact) in $Y$, hence the entire space since $Y$ is connected.
I'm wondering if there is a proof that does not make use of the rather sophisticated machinery of invariance of domain. The closest I can get is that if $f: X \to Y$ is a smooth embedding, then $f(X)$ must be open by the local submersion theorem. However, I can't see a way to transfer this to the topological category.
Thank you!