# Topology/Vector Bundles

A vector bundle is, broadly speaking, a family of vector spaces 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 ${\displaystyle B}$ is a space ${\displaystyle E}$ together with a continuous map ${\displaystyle p:E\rightarrow B}$ with the following properties:

(1) For each ${\displaystyle b\in B}$, ${\displaystyle p^{-1}(b)}$ is isomorphic to ${\displaystyle \mathbf {R} ^{n}}$

(2) ${\displaystyle B}$ is covered by open sets ${\displaystyle U_{i}}$ such that there exist homeomorphisms ${\displaystyle h_{i}:p^{-1}(U_{i})\rightarrow U_{i}\times \mathbf {R} ^{n}}$ and ${\displaystyle h_{i}\circ h_{j}^{-1}U_{i}\cap U_{j}\times \mathbf {R} ^{n}\rightarrow U_{i}\cap U_{j}\times \mathbf {R} ^{n}}$ is the identity on the first factor and a linear isomorphism on the second.

Replacing ${\displaystyle \mathbf {R} }$ with ${\displaystyle \mathbf {C} }$, we get the definition of a complex vector bundle.

We call ${\displaystyle E}$ the total space of the vector bundle and ${\displaystyle B}$, the base space.

One can define a smooth vector bundle as following:

${\displaystyle E}$ and ${\displaystyle B}$ 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.