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Well kept secret: all topologies are the Zariski topology
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if X is a Tychonoff space then the inclusion of X in the spectrum of its ring C(X) of continuous real-valued functions sending p to m_p = {f ∈ C(X): f(p) = 0} is a homeomorphism onto its image - the topology on X is a restriction of a Zariski topology! 2/n
Here, Tychonoff means both Hausdorff and Completely Regular. In particular, every topological manifold is Tychonoff, so this covers a hell of a lot of spaces. But why is it true? Well... 3/n
'Hausdorff' is there to purely make sure the map is injective*, so that's all good. But where did completely regular come from, and why does that make it work? 4/n
Well completely regular,by definition, means you can always separate a point from a closed set by means of a continuous function vanishing on the closed set but not at the point 5/n
But this means that every closed set C can be written as the intersection of a family of zero sets of cts functions - for each point p not in C, pick a function f_p vanishing on C but not on that point. Then C=∩_p f^{-1}(0) 6/n
So if I is the ideal generated by the f_p, then C is the set of common zeroes of the ideal I. Since C was just any closed set, this shows that the closed sets of a completely regular space are precisely the zero sets of ideals of continuous functions 7/n
So, if we restrict to the image of X in Spec C(X), the closed sets coincide! So most reasonable topologies are actually zariski topologies 8/8
