[proofplan] We prove the result by unpacking the definitions. A diffeomorphism is a bijective smooth map whose inverse map is also smooth. Since smooth maps between smooth manifolds are continuous with respect to the manifold topologies, both $F$ and its inverse are continuous. The definition of homeomorphism then gives the conclusion. [/proofplan]

[step:Unpack the diffeomorphism hypothesis]Since $F: M \to N$ is a diffeomorphism, by definition $F$ is bijective and smooth, and its inverse map $F^{-1}: N \to M$ is smooth. Let $G: N \to M$ denote this inverse map, so $G = F^{-1}$. Let $\operatorname{id}_M: M \to M$ and $\operatorname{id}_N: N \to N$ denote the identity maps on $M$ and $N$, respectively. Since $G$ is the inverse of $F$, \begin{align*} G \circ F = \operatorname{id}_M \end{align*} and \begin{align*} F \circ G = \operatorname{id}_N. \end{align*}[/step]

[guided]The hypothesis says more than just that $F$ is a smooth map. By the definition of a diffeomorphism, $F: M \to N$ is a bijection, $F$ is smooth, and the inverse function is also smooth. We denote this inverse by $G: N \to M$, so $G(q)$ is the unique point $p \in M$ such that $F(p) = q$. Let $\operatorname{id}_M: M \to M$ and $\operatorname{id}_N: N \to N$ denote the identity maps on $M$ and $N$, respectively. Because $G$ is the inverse of $F$, the two compositions are identity maps: \begin{align*} G \circ F = \operatorname{id}_M \end{align*} and \begin{align*} F \circ G = \operatorname{id}_N. \end{align*} These identities record exactly that $G$ is a two-sided inverse, which is the inverse map required in the definition of a homeomorphism.[/guided]

[step:Use smoothness to obtain continuity of $F$]A smooth map between smooth manifolds is continuous with respect to the underlying manifold topologies. Since $F: M \to N$ is smooth, it follows that $F$ is continuous as a map of topological spaces.[/step]

[guided]The next point is where smooth structure gives topological information. A smooth map between smooth manifolds is continuous with respect to the manifold topologies, because smoothness is defined by requiring the coordinate representatives in charts to be smooth maps between open subsets of Euclidean spaces, and Euclidean smooth maps are continuous. The hypothesis from the preceding step gives that $F: M \to N$ is smooth. Therefore $F$ is continuous when $M$ and $N$ are regarded only as topological spaces with their underlying manifold topologies.[/guided]

[step:Use smoothness of the inverse to obtain continuity of $F^{-1}$]The inverse map $G = F^{-1}: N \to M$ is smooth by the diffeomorphism hypothesis. Applying again the fact that smooth maps between smooth manifolds are continuous, $G$ is continuous as a map of topological spaces. Thus $F^{-1}: N \to M$ is continuous.[/step]

[guided]A homeomorphism requires continuity in both directions, so continuity of $F$ alone is not enough. The diffeomorphism hypothesis also states that the inverse function $G = F^{-1}: N \to M$ is smooth. Applying the same standard fact that smooth maps between smooth manifolds are continuous with respect to the manifold topologies, $G$ is continuous as a map from the underlying [topological space](/page/Topological%20Space) of $N$ to the underlying topological space of $M$. Since $G = F^{-1}$, this proves that $F^{-1}: N \to M$ is continuous.[/guided]

[step:Conclude that $F$ is a homeomorphism]We have shown that $F: M \to N$ is bijective, continuous, and has continuous inverse $F^{-1}: N \to M$. By the definition of a homeomorphism, $F$ is a homeomorphism between the underlying topological spaces of $M$ and $N$.[/step]

[guided]We now compare exactly what has been proved with the definition of a homeomorphism. A map between topological spaces is a homeomorphism when it is bijective, continuous, and its inverse map is continuous. From the diffeomorphism hypothesis, $F: M \to N$ is bijective. From the continuity step for smooth maps, $F$ is continuous. From the continuity step applied to the smooth inverse $F^{-1}: N \to M$, the inverse map is continuous. These three properties are precisely the definition of a homeomorphism between the underlying topological spaces of $M$ and $N$.[/guided]