as with the tensor product of two vector spaces introduced above. In general, there are no relations between and Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing yields the exterior algebra. When a field, is explicitly stated, a common term used is -algebra. Related structures Vector bundles A vector bundle is a family of vector spaces parametrized continuously by a topological space X. More precisely, a vector bundle over X is a topological space E equipped with a continuous map such that for every x in X, the fiber π−1(x) is a vector space. The case dim is called a line bundle. For any vector space V, the projection makes the product into a "trivial" vector bundle. Vector bundles over X are required to be locally a product of X and some (fixed) vector space V: for every x in X, there is a neighborhood U of x such that the restriction of π to π−1(U) is isomorphic to the trivial bundle . Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space X) be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle ). For example, the Möbius strip can be seen as a line bundle over the circle S1 (by identifying open intervals with the real line). It is, however, different from the cylinder , because the latter is orientable whereas the former is not. Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S1 is globally isomorphic to , since there is a global nonzero vector field on S1. In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S2 which is everywhere nonzero. K-theory studies the isomorphism classes of all vector bundles over some topological space. In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions O. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms. Modules