In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset.
Taking the complement of a single point in projective space of dimension at least 2 gives a non-affine quasi-projective variety.
is not closed since any polynomial zero on the complement must be zero on the affine line.
For another example, the complement of any conic in projective space of dimension 2 is affine.
This yields a basis of affine sets for the Zariski topology on a quasi-projective variety.