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?