[Math] An injective immersion is an embedding if and only if open onto image if and only if image is (regular) submanifold

Question

Update: Tim kinsella appears to say yes here.

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Related:

1. An injective smooth function with injective differential must have a continuous inverse?

2. I think my question is equivalent to the converse of this.

Let $N$ and $M$ be smooth manifolds with dimensions. Let $p \in N$. Let $F: N \to M$ be a smooth map.

A. $F$ is said to be an embedding if $F$ is an immersion and a topological embedding. (You may have a different but equivalent definition.)

B. Let $X$ and $Y$ be topological spaces. A map, not necessarily injective or continuous, $F: X \to Y$ is said to be a topological embedding if $\tilde F: X \to F(X)$ is a homemorphism.

C. $F$ is said to be open onto its image if $\tilde F$ is open.

D. If $F$ is an injective immersion, then $\tilde F$ is already injective ($\tilde F$ is injective if and only if $F$ is too), surjective and continuous (because $F$ is continuous because $F$ is smooth). Therefore, $F$ is an embedding if and only if $F$ is open onto its image.

Question: Is $\tilde F$ open if and only if $F$ is an embedding if and only if $F(N)$ is a regular submanifold (with dimension) of $M$? I think yes. Please verify.

Suppose $\tilde F$ open. Show $F(N)$ is a regular submanifold of $M$: Done here (which assumes open onto image rather than open).

Suppose $F(N)$ is a regular submanifold of $M$. Show $\tilde F$ is open.

1. Because $F(N)$ is a regular submanifold of $M$, $F(N)$ is a manifold, so $\tilde F$ is a map of manifolds, so we can talk about $\tilde F$ as smooth or not smooth. Unsurprisingly, because $F(N)$ is a regular submanifold of $M$, $\tilde F$ is smooth by this. Thus, we can talk about $\tilde F$ possibly being an immersion, submersion, local diffeomorphism, embedding, diffeomorphism, etc.

2. $\tilde F$ is an immersion if and only if $F$ is an immersion, by (1) and this,

3. $\tilde F$ is a surjective immersion, by (2) and (D),

4. $\dim F(N) = \dim N$, by (3) and this (or by (3) and this, I guess; Maybe the two arguments, one of using this and one of using this are actually equivalent).

5. $\tilde F$ is both a (surjective) submersion and a (surjective) local diffeomorphism, by (4) and this.

6. Both submersions and local diffeomorphisms are open maps.

7. By (5) and (6), $\tilde F$ is open.

Answer 1

$\tilde{F}: N \rightarrow F(N)$ is a smooth bijection with non degenerate derivative between two manifolds. so it's a diffeomorphism. in particular, it's an open map.