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I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group).

For any topological space, $X$, we can define $\mathcal O_X :=\{f:X\to \mathbb C |f \text { is continuous} \}$, which is a ring with pointwise addition and multiplication. Then we can look at the affine scheme $\text {Spec}_{\mathcal O_X}$.

My question is: can we always recover the space $X$ from this scheme? I feel like the answer should be no, because there can be some really weird topological spaces. (I know my question is fairly vague, but I hope it can be received well)

Matthew Levy
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  • A little unrelated, but if $X$ is compact Hausdorff, $C(X)$ is ring of real functions on $X$, then one can recover $X$ as the maximal spectrum $\text{mSpec} C(X)$. Indeed, there is a natural homeomorphism $X \to \text{mSpec} C(X)$ given by $x \mapsto \mathfrak{m}_x$ where $\mathfrak{m}_x \subset C(X)$ is the maximal ideal consisting of all continuous functions vanishing on $x$, the topology on the maximal spectrum being the subspace topology inherited from the Zariski topology on the prime spectrum. [ref : Atiyah-MacDonald, the penultimate exercise on chapter 1, iirc] – Balarka Sen Aug 22 '15 at 17:26
  • Additionally, I want to point out that this thing also holds for affine varieties $V$ and maximal spectrum $\text{mSpec} , k[V]$ of it's coordinate ring, where the variety is given the Zariski topology - it's a weak form of Nullstellensatz. Note that in both cases, we consider maximal spectrum instead of prime spectrum. This is because the prime spectrum holds way too much data. – Balarka Sen Aug 22 '15 at 17:35
  • If you want an analogue of this to work for prime spectrums, the correct statement is obtained by replacing affine varieties by affine schemes (in which case the fact is tautological, as ring of functions on $\text{Spec} , R$ is just $R$) – Balarka Sen Aug 22 '15 at 17:35

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The answer is no in general, because if $X$ is any nonempty set with the indiscrete topology (only $\emptyset$ and itself are open) then $\mathcal{O}_X \cong \mathbb{C}$, while two indiscrete spaces of different cardinality are not homeomorphic.

When you restrict to compact Hausdorff spaces, this becomes true. (See A theorem due to Gelfand and Kolmogorov for the case of real-valued functions). There are several strong correspondences between compact Hausdorff spaces (even locally compact) and their functions into $\mathbb{C}$ and $\mathbb{R}$. A keyword is Gelfand duality.

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