It is a kind of folklore but I would like to see the proof of the following fact: given two smooth manifolds $M$ and $N$ if we assume that the algebras $C^{\infty}\_0(M)$ and $C^{\infty}\_0(N)$ are isomorphic (as algebras) then $M$ and $N$ are diffeomorphic. 



Isomorphic algebras determine diffeomorphic manifolds
Theorem 2.8 on p. 30 of *Models for smooth infinitesimal analysis* by Moerdijk and Reyes. The theorem works with $C^\infty$ rings, which are a bit more general than algebras of functions. The advantage is that the proof is short.


