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.