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 S is globally isomorphic to S × R, since there is a global nonzero vector field on S. In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S 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.