Topology/Vector Bundles
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A vector bundle is, broadly speaking, a family of vectors bundles 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 B is a space E together with a continuous map 
with the following properties:
(1) For each 
, p − 1(b) is isomorphic to 
(2) B is covered by open sets Ui 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 E the total space of the vector bundle and B, the base space.
One can define a smooth vector bundle as following:
E and B 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.
