Proof that we have a differentiable manifold structure

Question

If I find a diffeomorphism between two sets $M$ and $N$ such that $M$ is a manifold, does that imply that $N$ is a manifold too?

Answer 1

In order to talk about diffeomorphisms $f:M\rightarrow N$, you need to fix a smooth structure on $M$ and $N$. So you need to know already that $M$ and $N$ admit a smooth manifold structure. However, you can do the following.

Say $M$ is a manifold with smooth structure $\mathcal{A}=\{(U_{i},\phi_{i})\}$. Let $N$ be just a set, and assume $f:M\rightarrow N$ is a bijection. Then you can give $N$ a smooth manifold structure as follows:

1) Transport the topology of $M$ to $N$. That is, we endow $N$ with the unique topology that turns $f$ into a homeomorphism. Namely, define the open subsets of $N$ to be the images under $f$ of the open subsets of $M$.

2) Transport the smooth structure $\mathcal{A}$ of $M$ to $N$. That is, we endow $N$ with the unique smooth structure $\mathcal{A}_{f}$ that turns $f$ into a diffeomorphism. Namely, we define $\mathcal{A}_{f}:=\{(f(U_{i}),\phi_{i}\circ f^{-1})\}$.