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I'm studying Fulton's book and in the chapter $6$, he gives a definition of variety. Before that he defines the affine variety in the chapter 2:

The definition of affine variety is

The definition of variety:

This definition of variety is a generalization of the concept of affine variety? I couldn't see why.

Thanks in advance

user42912
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    Which part of being a generalization do you not see? The part about an affine variety being a variety, or that there are non-affine varieties? – Tobias Kildetoft Sep 07 '13 at 18:30
  • @TobiasKildetoft An affine variety $K$ is a variety because we can take $V=K$, am I right? – user42912 Sep 07 '13 at 18:39
  • Not quite. For example, the $V$ in the definition of variety could be projective. – Tobias Kildetoft Sep 07 '13 at 18:42
  • @TobiasKildetoft you're right, so I don't see any two parts of the generalization. – user42912 Sep 07 '13 at 18:45
  • @TobiasKildetoft wait, I don't understand, in particular $V$ could be a set in $\mathbb A^n$ – user42912 Sep 07 '13 at 18:48
  • Yes, it could. But in the given definition, $V$ could be many things. You can certainly pick things in such a way that $K$ and $V$ coincide, but you need to pick things correctly. My main objection to your first comment is that $V$ is not really something fixed in this case that it makes sens to set $K$ equal to. – Tobias Kildetoft Sep 07 '13 at 18:50
  • @TobiasKildetoft I think I understood your point, $V$ is not fixed. In this way, any irreducible algebraic subset $K$ of $\mathbb A^n$ is a variety because it's open in the induced topology on $K$. Am I right? – user42912 Sep 07 '13 at 19:06

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