In Gillman and Jerrison's book, "Rings of Continuous Functions", they show a nice relationship between the set of z-ideals in C(X) and the set of filters on X. One can go for with this relationship; for example, proving the smallest measurable cardinal (if any exist) is strongly inaccessible, seems to be better understood from a filter perspective (rather than ideal).

Now instead of considering the spectrum of a ring, is there any usefulness to consider the set of filters (which consist of algebraic sets), and defining a topology on it? It just seems like some things (like analytic stuff) would be more natural working in a filter setting.

I suppose for a variety V, you could work with something like Filt(V) instead of Spec(V). Where Filt(V) consists of fixed filters of a certain type (each filter would correspond to a variety). Is there any study of this kind? Or is this analogy not very useful?

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