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Recently I was thinking about the following:

When talking about the sphere $\mathbb{S}^1$ we can topologize it in two different ways. Either as a subspace of the euclidean space $\mathbb{R}^2$ or as a closed (real) algebraic subvariety of $\mathbb{A}^2_\mathbb{R}$. In either case we think about it as the same "thing", namely the points $(x,y)\in\mathbb{R}^2$ such that $x^2+y^2=1$.

As algebraic geometry loves algebraically closed sets we can also consider the setup in $\mathbb{C}^2$ and note, that $\lbrace(x,y)\in\mathbb{C}^2\vert x^2+y^2=1\rbrace$ can topologized either with the euclidean topology or the Zariski topology, and although these two versions are different as topological spaces, they are the same "thing". Of course we don't need to restrict ourselves to this example, but can consider any affine algebraic variety over $\mathbb{C}$ (or even any subset of $\mathbb{C}^n$) with either the Zariski topology or the euclidean topology and feel that they are the same "thing".

Is there any way to make this idea formal? Or is it maybe even possible to reconstruct the euclidean topology directly from the Zariski topology?

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    $\begingroup$ I don't think so: Zariski topology is not Hausdorff, you can't hope whatever. $\endgroup$ – Bernard Dec 27 '17 at 0:00
  • $\begingroup$ The same "thing" is just the same set of points, with no topology on it. $\endgroup$ – Javi Dec 27 '17 at 0:08
  • $\begingroup$ Yeah the whole point of algebraic geometry is not to use euclidean topologies. You’re making Groethendieck spin. $\endgroup$ – Randall Dec 27 '17 at 0:13
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    $\begingroup$ That is not "the whole point of algebraic geometry" by a faaaaaaaaaar stretch. $\endgroup$ – Mariano Suárez-Álvarez Dec 27 '17 at 0:56
  • $\begingroup$ The topologies alone are far too weak a structure to do anything useful with, but it sounds like what you really are looking for is GAGA. $\endgroup$ – Eric Wofsey Dec 27 '17 at 1:48
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They are certainly not the same thing, but they are related. There is an analytification functor $X \mapsto X^{an}$ sending complex varieties to complex analytic spaces, a generalization of complex manifolds; these have the Euclidean / analytic topology but are also equipped with a sheaf of holomorphic functions. The study of the relationship between $X$ and $X^{an}$ goes by the name "GAGA".

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  • $\begingroup$ Thank you very much for your answer. Is there also something similar for $\mathbb{R}$? $\endgroup$ – Takirion Jan 6 '18 at 11:15
  • $\begingroup$ You can consider the functor sending real varieties to their real points with the analytic topology but it's much less well behaved; for starters it isn't faithful. See en.wikipedia.org/wiki/Real_algebraic_geometry. The whole subject has a very different flavor. $\endgroup$ – Qiaochu Yuan Jan 6 '18 at 19:53

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