For any topological space $X$, we can define $C(X)$ to be the commutative ring of continuous functions $f\,:\,X\rightarrow \mathbb{R}$ under pointwise addition and multiplication. Then $C(-)$ becomes a contravariant functor $C(-)\,:\,\bf{Top}\rightarrow \text{ComRing}$.
A theorem due to Gelfand and Kolmogorov states the following:
Let $X$ and $Y$ be compact Hausdorff spaces. If $C(X)$ and $C(Y)$ are isomorphic as rings, then $X$ and $Y$ are homeomorphic.
I encountered this theorem as an example in a book on homological algebra, without proof. I have searched for the proof, but have been unable to find it.
If anyone has an idea of how to prove this, or a reference to a proof, I would appreciate it greatly.