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.