have been found; for example, it is straightforward to construct a toric variety that is not quasi-projective but complete. Examples Subvariety A subvariety is a subset of a variety that is itself a variety (with respect to the structure induced from the ambient variety). For example, every open subset of a variety is a variety. See also closed immersion. Hilbert's Nullstellensatz says that closed subvarieties of an affine or projective variety are in one-to-one correspondence with the prime ideals or irrelevant homogeneous prime ideals of the coordinate ring of the variety. Affine variety Example 1 Let , and A2 be the two-dimensional affine space over C. Polynomials in the ring C[x, y] can be viewed as complex valued functions on A2 by evaluating at the points in A2. Let subset S of C[x, y] contain a single element : The zero-locus of is the set of points in A2 on which this function vanishes: it is the set of all pairs of complex numbers (x, y) such that y = 1 − x. This is called a line in the affine plane. (In the classical topology coming from the topology on the complex numbers, a complex line is a real manifold of dimension two.) This is the set : Thus the subset of A2 is an algebraic set. The set V is not empty. It is irreducible, as it cannot be written as the union of two proper algebraic subsets. Thus it is an affine algebraic variety. Example 2 Let , and A2 be the two-dimensional affine space over C. Polynomials in the ring C[x, y] can be viewed as complex valued functions on A2 by evaluating at the points in A2. Let subset S of C[x, y] contain a single element g(x, y): The zero-locus of g(x, y) is the set of points in A2 on which this function vanishes, that is the set of points (x,y) such that x2 + y2 = 1. As g(x, y) is an absolutely irreducible polynomial, this is an algebraic variety. The set of its real points (that is the points for which x and y are real numbers), is known as the unit circle; this name is also often given to the whole variety. Example 3 The following example is neither a hypersurface, nor a linear space, nor a single point. Let A3 be the three-dimensional affine space over C. The set