Is open (topological) smooth embedding equivalent to injective local (homeomorphism) diffeomorphism

Question

I could have sworn I saw a question like this before, but I can't find it. Anyway:

Let $F: N \to M$ be a (continuous) smooth map from (topological) smooth $n$-manifold $N$ to (topological) smooth $m$-manifold $M$.

Question 1: Is this true?

$F$ is open (topological) smooth embedding if and only if injective local (homeomorphism) diffeomorphism.

Question 2 and Context for Question 1: $A$ is a smooth embedded $k$-submanifold of $M$, with $0 \le k \le m$, if and only if inclusion $\iota: A \to M$ is a smooth embedding. (This is either a theorem or a definition.) When we have $k=m$ itself, $\iota$ must surely be more than just a smooth embedding. I believe $\iota$ is an open smooth embedding, equivalently an injective local diffeomorphism. Is this right? And then for continuous case: upgrade topological embedding to open topological embedding, i.e. injective local homeomorphism. (See here or here.)

Definitions, Notes (I moved proof to answer):

• Definition of topological embedding := The induced map $\tilde F: N \to F(N)$ is a homeomorphism = $F$ is injective, continuous and open onto its image. = $\tilde F$ is injective, continuous and open.

• Definition 1 of smooth embedding := topological embedding + immersion

• Definition 2 of smooth embedding: $F(N)$ is smooth embedded $m$-submanifold of $M$, and then the induced map $\tilde F: N \to F(N)$ is diffeomorphism. (or say $F(N)$ is smooth embedded $k$-submanifold with $0 \le k \le m$ and then $k$ will turn out $k=m$.)

• Definition of local (homeo) diffeo: For all $p \in N$, there exists open subset $p \in U$ of $N$ s.t. $F(U)$ is open in $M$ and $F|_U: U \to F(U)$ is a (homeomorphism) diffeomorphism.

  • Note 1: By '$F|_U: U \to F(U)$', I of course mean: Let $H=F|_U: U \to M$. Then consider the induced map $G=\tilde H: U \to H(U)=F(U)$. Then '$F|_U: U \to F(U)$' refers to $G$ instead of $H$.

  • Note 2: I understand '$F|_U: U \to F(U)$ is a diffeomorphism.' to be equivalent to '$F|_U: U \to M$ is a smooth embedding.' (In re Note 1: $G$ is diffeomorphism if and only if $H$ is smooth embedding.)

  • Note 3: I understand the following conditions to be equivalent:

    • Condition 3.1. $F(U)$ is open in $M$.
    • Condition 3.2. $F(U)$ is a smooth $k$-submanifold of $M$, and $n=m$.
    • Condition 3.3. $F(U)$ is a smooth $k$-submanifold of $M$, and $F(N)$ is open in $M$.
    • Condition 3.4. $F(N)$ is smooth $k$-submanifold of $M$, and then $F(U)$ is open in $F(N)$.

Answer 1

For continuous case:

• If direction:

Local homeo is open onto its image: Open (see (1)) is actually stronger than open onto its image (and is equivalent to open onto its image + its image is open).

• Only if direction:

Open topological embedding is local homeo: Let $p \in N$. Choose $U = N$ itself.

For smooth case:

• If direction (using Definition 1):

1. Local (homeo) diffeo is open.
2. Local diffeo is immersion. (actually immersion = local smooth embedding. So, given that diffeo implies smooth embedding, we can think of adding 'local' to both sides of the implication.)
3. (For continuous case)
4. Injective local diffeo is (injective local homeo and thus a fortiori) a topological embedding.

• If direction (using Definition 2):

1. Local diffeo is open.
2. By (1), image is (topological) smooth embedded $m$-manifold of $M$. (The conclusion here is equivalent to that $\iota: F(N) \to M$ is a smooth embedding. However, we still wanna show $F$ is also a smooth embedding.)
3. $\tilde F: N \to F(N)$ is diffeo:

• 3.1. $\tilde F$ is open if and only if $F$ is open onto its image: '$F$ is open' is stronger than '$F$ is open onto its image' (and is equivalent to open onto its image + its image is open).

• 3.2. $\tilde F$ is injective since $F$ is injective.

• 3.3. $\tilde F$ is smooth: It's a rule that $F$ and $F(N)$ smooth $k$-submanifold (even if $k \ne m$) implies $\tilde F$ smooth.

• 3.4. $\tilde F^{-1}$ is smooth: It's a rule that this is equivalent to that $F$ is an immersion. It's part of proving the equivalence of Definitions 1 and 2.

• Only if direction:

1. Smooth embedding is (topological embedding and thus a fortiori) injective.
2. Open smooth embedding is local diffeo: Let $p \in N$. Choose $U = N$ itself.