# Does every quasi-affine variety have an open cover of affine dense subsets?

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?

• Seems good to me. – Amitai Yuval Nov 16 '14 at 22:17

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.