Is it true that for either complex or real differentiable manifolds, the space of points of the manifolds corresponds to the maximal (or prime) spectrum of the ring of functions (holomorphic functions in the complex case, real-valued differentiable functions in the real case) on the manifold? I'm curious how similar to algebraic geometry these cases are.
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https://math.stackexchange.com/questions/1406035/spectrum-of-the-ring-of-continuous-functions-on-a-space and https://math.stackexchange.com/questions/226736/a-theorem-due-to-gelfand-and-kolmogorov may be useful. Briefly, the maximal spectrum of the ring of continuous real-valued functions on a compact Hausdorff space is naturally homeomorphic to the space. For complex-analytic manifolds, the only chance for any homeomorphism is if your manifold is Stein, but I do not know the answer in this case. – KReiser Aug 01 '18 at 04:24