Known fact: If $X$ is a compact, Hausdorff space, then $X$ is homeomorphic to the max spectrum of $C(X)$ with the Zariski topology.
This fails for non-compact spaces, as for instance there may be too many maximal ideals.
I thought about the example of $\mathbb{R}$ for a while, and found that one difference in this case is the ideal of compactly supported functions is a proper ideal, which thus contained in many maximal ideals $m$ not of the form "ideal of functions vanishing at a given point $p \in X$." I was unable to determine the field $C(R)/m$ for one of these maximal ideals. Is it not $\mathbb{R}$? What can they be?
More general question: If $X$ is a non-compact real manifold, then is $X$ homeomorphic to the real points of $\operatorname{mSpec}( C(X))$, where the real points means the maximal ideals whose residue field gives a trivial extension of $\mathbb{R}$?
C(X) refers to the ring of continuous functions on $X$.