Examples
Since quasiprojective varieties generalize both affine and projective varieties, they are sometimes referred to simply as varieties. Varieties isomorphic to affine algebraic varieties as quasiprojective varieties are called affine varieties; similarly for projective varieties. For example, the complement of a point in the affine line, i.e., is isomorphic to the zero set of the polynomial in the affine plane. As an affine set X 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. Varieties isomorphic to open subsets of affine varieties are called quasi-affine.
Quasiprojective varieties are locally affine in the sense that a manifold is locally Euclidean — every point of a quasiprojective variety has a neighborhood given by an affine variety. This yields a basis of affine sets for the Zariski topology on a quasiprojective variety.
Read more about this topic: Quasiprojective Variety
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