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Suppose you have a nonempty, quasi-affine variety $Y$. Does $Y$ always have an open cover of affine dense subsets?

I know that every quasi-affine variety has an open cover by quasi-affine varieties which are isomorphic to affine varieties. Also, I know that any nonempty open subset of an affine variety is always dense.

Is it correct to conclude that these together imply that $Y$ has an open cover of affine dense subsets, or am I missing some subtlety?

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  • $\begingroup$ Seems good to me. $\endgroup$ – Amitai Yuval Nov 16 '14 at 22:17
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Any variety $Y$ has a cover by open affines, and these are automatically dense in $Y$ since any non-empty open subset of an irreducible topological space is dense.
Quasi-affineness is irrelevant to this question: only the irreducibility of $Y$ plays a role.

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  • $\begingroup$ Thank you Georges. Sorry to come back to this so late, but do you know if this still is true if we don't include irreducibility in the definition of an affine variety? $\endgroup$ – Lena Richman Feb 1 '15 at 8:20

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