Uniqueness of topology and smooth structure on an immersed submanifold

Solution 1:

The concept of an immersed manifold often leads to confusion and I do not see a real benefit to use it.

Let us consider a more general situation: Given a smooth manifold $M$, a set $S$ and a bijection $f : N \to S$. Then it is clear that

1. there exists a unique topology $\tau$ on $S$ making $f : N \to (S,\tau)$ a homeomorphism.

2. there exists a unique smooth structure $\mathfrak D$ on $(S,\tau)$ making $f : N \to (S,\tau,\mathfrak D)$ a diffeomorphism.

Lee uses this with $S = F(N)$. His definition of open sets and of charts is nothing else than the explicit construction of $\tau$ and $\mathfrak D$.

The problem is that $\tau$ in general is not the subspace topology inherited from $M$. As an example take the injective smooth immersion $\beta : N = (-\pi,\pi) \to \mathbb R^2$ decribed here. Its image $S$ is a "figure eight" which is compact and thus cannot be homeomorphic to $N$. Moreover $S$ is not a manifold. Thus, unfortunately, your approach does not work.

You see that the "immersed submanifold structure" of $S$ is a fairly artificial thing which has nothing to do with the subspace $S$ of $\mathbb R^2$.

Also have a look at Restricting the codomain of a smooth map to submanifold is smooth.

Update:

Let us consider the bijective function $\bar F : N \stackrel{F}{\to} F(N)$. We endow $F(N)$ with the unique topology and smooth structure making $\bar F$ a diffeomorphism. If $\iota : F(N) \hookrightarrow M$ denotes inclusion, then clearly $\iota \circ \bar F = F$. Since $\bar F$ is a diffeomorphism, we get $\iota = F \circ \bar F^{-1}$. This shows that $\iota$ is an injective immersion because $F$ is one.