Topology/Vector Bundles

A vector bundle is, broadly speaking, a family of vector spaces which is continuously indexed by a topological space. An important example is the tangent bundle of a manifold. The formal definition is the following: Definition (Vector bundle) A real vector bundle on a topological space is a space together with a continuous map with the following properties: (1) For each , is isomorphic to (2) is covered by open sets such that there exist homeomorphisms and is the identity on the first factor and a linear isomorphism on the second. Replacing with , we get the definition of a complex vector bundle. We call the total space of the vector bundle and , the base space. One can define a smooth vector bundle as following: and have to be smooth manifolds and every maps appearing in the previous definition have to be smooth. As we have stated before, the tangent bundle of a smooth manifold is a smooth vector bundle.