Uniqueness of smooth structure on a set The question states that if $M$ is a manifold and $B$ be a set and $F:M \to B$ is a bijection then to show there exists a unique smooth structure on $B$ that $F$ is a diffeomorphism.
I am really not sure how to get a smooth structure on $B$ using the bijective ness of the given function $F$ also how to get the uniqueness. Any hints regarding how to proceed would be very much helpful.
 A: First of all you have to transfer on $B$ the topology structure:
$\tau_B:=\{ V: F^{-1}(U) \in \tau\}$
is a topology on $B$, where $\tau$ is the topology on $M$. This is the quotient topology of the map $F$ on $B$ and it’s the biggest topology such that the map is continuos. In our case $F$ becomes an homeomorphism.
This permit us to say that $B$ is Hausdorff and second countable.
Now you have to define an atlas on $B$:
$\mathcal{A}_B:=\{(F(U), \phi_U\circ F^{-1})\}_{\{(U,\phi_U)\in \mathcal{A}\}}$
With respect this atlas $B$ is a manifold and $F$ becomes a diffeomorphism.
General fact:
Given a manifold $N$ with two differentiable structures $\mathcal{A}$ and $\mathcal{A}’$, then they are equivalent if and only if there exists a diffeomorphism
$G: (N, \mathcal{A}) \to (N, \mathcal{A}’)$
This permit us to say that $\mathcal{A}_B$ is the only structure on $B$ such that $F$ is a diffeomorphism. In fact if it’s not so, then you can consider
$F\colon (M, \mathcal{A})\to (B, \mathcal{A}_B’)$
And
$F^{-1}\colon (B, \mathcal{A}_B)\to (M,\mathcal{A})$
To have the identity diffeomorphism $F\circ F^{-1}$ and conclude that $\mathcal{A}_B’$ and $\mathcal{A}_B$ are equivalent structures.
