Over a small neighborhood of a manifold, a vector bundle is spanned by the local sections defined on . For example, in a coordinate chart with coordinates , every smooth vector field can be written as a sum where are smooth functions. The vector fields span the space of vector fields, considered as a module over the ring of smooth real-valued functions. On this coordinate chart , the tangent bundle can be written . This is a trivialization of the tangent bundle.
In general, a vector bundle of bundle rank is spanned locally by independent bundle sections. Every point has a neighborhood and sections defined on , such that over every point in the fibers are spanned by those sections.
Similarly, for a fiber bundle, near every point , there is a neighborhood such that the bundle over is , where is the fiber.
A bundle is a set of trivializations that cover the base manifold. The trivializations are put together to form a bundle with its transition functions.