arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows. Let K be any division ring (skewfield). Let K3 denote the set of all triples x = of elements of K (a Cartesian product viewed as a vector space). For any nonzero x in K3, the minimal subspace of K3 containing x (which may be visualized as all the vectors in a line through the origin) is the subset of K3. Similarly, let x and y be linearly independent elements of K3, meaning that implies that . The minimal subspace of K3 containing x and y (which may be visualized as all the vectors in a plane through the origin) is the subset of K3. This 2-dimensional subspace contains various 1-dimensional subspaces through the origin that may be obtained by fixing k and m and taking the multiples of the resulting vector. Different choices of k and m that are in the same ratio will give the same line. The projective plane over K, denoted PG(2, K) or KP2, has a set of points consisting of all the 1-dimensional subspaces in K3. A subset L of the points of PG(2, K) is a line in PG(2, K) if there exists a 2-dimensional subspace of K3 whose set of 1-dimensional subspaces is exactly L. Verifying that this construction produces a projective plane is usually left as a linear algebra exercise. An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the set modulo the equivalence relation x ~ kx, for all k in K×. Lines in the projective plane are defined exactly as above. The coordinates of a point in PG(2, K) are called homogeneous coordinates. Each triple represents a well-defined point in PG(2, K), except for the triple , which represents no point. Each point in PG(2, K), however, is represented by many triples. If K is a topological space, then KP2, inherits a topology via the product, subspace, and quotient topologies. Classical examples The real projective plane RP2 arises when K is taken to be the real numbers, R. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology. In this construction, consider the unit sphere centered at the origin in R3. Each of the R3 lines in this construction intersects the