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This might be a somewhat stupid question, but I've been wondering if it is possible to define some other topology on $\mathrm{Spec} (A)$ other than Zariski topology in a way that it has some interesting properties as well.

First of all, I am new as this is my first encounter with anything close or related to algebraic geometry, so be easy on me =).

And second, what I'd like to know is if, for example, there is a topology on $\mathrm{Spec} (A)$ such that, say, $\mathrm{Spec} (A)$ is Hausdorff or has any other nice properties (connectedness, compactness, etc...), or why if such a topology exists, isn't as interesting as Zariski topology.

Note: I am aware of a "similar" question here. However, I'm not interested that much in why Zariski topology is important since I think I understand how it arises naturally.

user313212
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2 Answers2

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As far as I'm aware there is no other interesting topology on $\mathrm{Spec} (A)$. However:

1) There are the so called "Grothendieck topologies" on any $\mathrm{Spec} (A)$ or more generally on an arbitrary scheme.
They are generalizations of the usual notion of a topology and compensate for the coarseness of the Zariski topology, in that they allow results analogous to those in algebraic topology: Gysin sequence, Künneth formula, Poincaré duality, Lefschetz fixed point formula,etc.
They were dreamed of by Weil and constructed by Grothendieck (with the help of Mike Artin and others), spurred by an insight of Serre's.
Their most spectacular achievement was the complete solution of the Weil conjectures by Grothendieck and Deligne.

2) For schemes $X$ of finite type over $\mathbb C$, for example $\mathrm{Spec} (A)$ with $A$ a finitely generated $\mathbb C$-algebra, Serre showed how to give the set $X^{cl}$ of closed points of $X$ a topology and a structure sheaf such that $X^{cl}$ becomes a complex analytic space.
There is then an exciting interplay (foreshadowed by Riemann, Chow and others) between the scheme $X$ and the analytic space $X^{cl}$, in which algebra, algebraic topology and analysis join forces, to our greatest delight.
Hodge theory is an egregious example of that alliance.

Georges Elencwajg
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  • Thank you very much for your answe! Even though many of the concepts you showed in your answer are out of my reach for the moment, they look intriguing enough to hope one day I'll be able to understand them :) – user313212 May 02 '17 at 16:01
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    You are welcome, dear user313212. And I'm sure you will soon be able to understand all this, ...and much more! – Georges Elencwajg May 02 '17 at 17:36
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There is a generalization of the notion of topology called a Grothendieck topology. Grothendieck has defined many examples of Grothendieck topolgies in the category of schemes for example the Etale topology which is one notion used in the proof of Weil conjectures.

https://en.wikipedia.org/wiki/Grothendieck_topology

Tsemo Aristide
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