# Homeomorphism wiht image and diffeomorphism with image

I was thinking of sufficient conditions for a map $F:X\to Y_1\times Y_2$ to be homeomorphism with its image. So I thought the following facts and proofs:

If $F_1:X\to Y_1$ and $F_2: X \to Y_2$ are continuous and $F_1$ is an homemorphism with its image, then $F:X\to Y_1\times Y_2, \quad x\mapsto(F_1(x), F_2(x))$ is an homemoprhism with its image.

Proof.

$\overline{F}:X\to F(X)$ si clearly continuous and bijective. Its inverse is $\overline{F}^{-1}:F(X)\to X, \quad (x,y)\mapsto F_1^{-1}(x)$ wich is continuous since $\overline{F}^{-1}= \overline{F_1}^{-1} \circ (pr_1)_{|F(X)}:F(X)\to F_1(X)\to X$.

Then i was thinking if I could "generalize" this to smooth maps.

So i thought:

If $F_1:X\to Y_1$ and $F_2: X \to Y_2$ are $C^\infty$ and $F_1$ is a differomorpshism with its image (assuming $F_1(X)$ embedded submanifold of $Y_1$), then $F:X\to Y_1\times Y_2, \quad x\mapsto(F_1(x), F_2(x))$ is a diffeomorphism with its image (assuming that $F(X)$ is an embedded submanifold of $Y_1\times Y_2$).

Proof. Since $F_1$ and $F_2$ are $C^\infty$ then $F$ is $C^\infty$. Since $F(X)$ is an embedded submanifold of $Y_1\times Y_2$ then also $\overline{F}:X\to F(X)$ is $C^\infty$. The map $pr_1:Y_1 \times Y_2\to Y_1$ is $C^\infty$, so also $pr_1:F(X)\to Y_1$ is $C^\infty$ so also $(pr_1)_{|F(X)}:F(X)\to F_1(X)$ is $C^\infty$. The map $\overline{F_1}:X\to F_1(X)$ is a diffeomorphism. Then $\overline{F}^{-1}=\overline{F_1}^{-1} \circ (pr_1)_{|F(X)}:F(X)\to X$ is $C^\infty$.

My questions:

1) Are these facts and proofs correct?

2) Are there other (frequently used and "useful") sufficient conditions for a map to be an heoomorphism with its image or differomorpshims with its image?

3) Are the hypothesis: $F(X)$ embedded submanifold of $Y_1\times Y_2$ and $F_1(X)$ embedded submanifold of $Y_1$ necesary?

1) Many categories admit products. In this case, morphisms $F_1: X \to Y_1$ and $F_2: X \to Y_2$, where $X, Y_1, Y_2$ are objects of the respective categories, automatically yield a morphism of that category into the product $Y_1 \times Y_2$. Since the projections are also maps in the respective category, the partial inverse argument of yours works in this generality.