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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.

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You might already know this, but there is a similar statement which is known as Gelfand duality. It deals with functions to $\mathbb{C}$ and commutative $C^*$-algebras though. – Sebastian Nov 1 '12 at 12:37
Hist. ref.: И. М. Гельфанд, А. Н. Колмогоров, “О кольцах непрерывных функций на топологических пространствах”, Докл. АН СССР, 22:1 (1939), 11–15 – Grigory M Nov 1 '12 at 12:39
An english translation of (part of) the paper referenced by @Grigory's (I hope...). – Raymond Manzoni Nov 1 '12 at 12:47
up vote 6 down vote accepted

I found a proof in these lecture notes by Garrido and Jaramillo. See Theorem 18. They also have historical references.

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Gillman-Jerison, Rings of continuous functions, Theorem 7.3.

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Dugundji's Topology has a very short, readable proof.

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Can you provide a more specific reference? – Ludolila Jan 24 '14 at 12:29

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