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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?

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The first fact and proof are correct. The second is as well. More statements concerning the matter are as follows:

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.

2) What immediately comes to mind is the inverse function theorem, which given injectivity and a nonsingular differential yields exactly what you want. Note that the inverse function theorem also holds on manifolds.

3) That depends what kind of a morphism you want. If you want a morphism of manifolds, then certainly you have to make sure that the domain of definition is also a manifold. Otherwise, you may need to restrict to categories which allow more "unorthodox" subobjects; for instance, the category of topological spaces.

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