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)
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.