In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E→ B with E and B sets. It is no longer true that the preimages must all look alike, unlike fiber bundles where the fibers must all be isomorphic (in the case of vector bundles) and homeomorphic. Definition A bundle is a triple where are sets and is a map. is called the total space is the base space of the bundle is the projection This definition of a bundle is quite unrestrictive. For instance, the empty function defines a bundle. Nonetheless it serves well to introduce the basic terminology, and every type of bundle has the basic ingredients of above with restrictions on and usually there is additional structure. For each is the fibre or fiber of the bundle over . A bundle is a subbundle of if and . A cross section is a map such that for each , that is, . Examples If and are smooth manifolds and is smooth, surjective and in addition a submersion, then the bundle is a fibered manifold. Here and in the following examples, the smoothness condition may be weakened to continuous or sharpened to analytic, or it could be anything reasonable, like continuously differentiable (), in between. If for each two points and in the base, the corresponding fibers and are homotopy equivalent, then the bundle is a fibration. If for each two points and in the base, the corresponding fibers and are homeomorphic, and in addition the bundle satisfies certain conditions of local triviality outlined in the pertaining linked articles, then the bundle is a fiber bundle. Usually there is additional structure , e.g. a group structure or a vector space structure, on the fibers besides a topology. Then is required that the homeomorphism is an isomorphism with respect to that structure, and the conditions of local triviality are sharpened accordingly. A principal bundle is a fiber bundle endowed with a right group action with certain properties. One example of a principal bundle is the frame bundle. If for each two points and in the base, the corresponding fibers and are vector spaces of the same dimension, then