Consider the problem asked in my smooth manifolds assignment:

If an open set $U \subseteq\mathbb{R}^n$ is diffeomorphic to an open set $V\subseteq \mathbb{R}^m$ then prove that n=m.

I think the question means that there exits a diffeomorphism f (say) from $\mathbb{R}^n \to \mathbb{R}^m$. Now, I thought of proving that such map f cannot be both one -one or onto.But I am unable to argue because f is not given.So, how should I try proving that if such map is both 1-1 and onto (assuming that it exists), m=n.

If there is some other way of proving that m=n, that is also welcome.

## Best Answer

Let $F : U \rightarrow V$ be a diffeomorphism and let $p \in U$. Then the derivative $DF_p : T_pU \rightarrow T_{F(p)}V$ is an

isomorphism of vector spaces. Since $T_pU$ is isomorphic to $\mathbb{R}^n$ and $T_{F(p)}V$ is isomorphic to $\mathbb{R}^m$, we must have that $n = m$.