How to know about $X$ from $C^{\infty}(X)$?

Question

Let $X$ be a subset of $\mathbb{R}^n$ and $C^{\infty}(X)$ be its $C^{\infty}$-ring.

I think it is possible to know some topological properties of $X$ by examining $C^{\infty}(X)$. Where can I read this kind of study?

Answer 1

This is more general than what you asked for, but for any locally Hausdorff $ X $ (which any reasonable subspace of $ \mathbb{R}^n $ should be) you can recover it completely from its ring of smooth functions $ C^\infty (X;\mathbb{C})$ by considering the space of symbols on it, i.e. the ring homomorphisms to $ \mathbb{C} $. You should consider these as evaluation at a point, recovering exactly all of $ X$ (one can also recover the topology of X in this way).

This follows from the Gelfand-Naimark theorem, which gives a duality between the category of locally compact Hausdorff spaces and the category of so-called commutative $C^*$-algebras. I do not know of any good references, but googling Gelfand-Naimark should help.