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fiber bundle
Let $F$ be a topological space and $G$ be a topological group which acts on $F$ on the left. A fiber bundle with fiber $F$ and structure group $G$ consists of the following data:

a topological space $B$ called the base space, a space $E$ called the total space and a continuous surjective map $\pi:E\to B$ called the projection of the bundle,

an open cover $\{U_{i}\}$ of $B$ along with a collection of continuous maps $\{\phi_{i}:\pi^{{1}}U_{i}\to F\}$ called local trivializations and

a collection of continuous maps $\{g_{{ij}}:U_{i}\cap U_{j}\to G\}$ called transition functions
which satisfy the following properties
1. the map $\pi^{{1}}U_{i}\to U_{i}\times F$ given by $e\mapsto(\pi(e),\phi_{i}(e))$ is a homeomorphism for each $i$,
2. for all indices $i,j$ and $e\in\pi^{{1}}(U_{i}\cap U_{j})$, $g_{{ji}}(\pi(e))\cdot\phi_{i}(e)=\phi_{j}(e)$ and
3. for all indices $i,j,k$ and $b\in U_{i}\cap U_{j}\cap U_{k}$, $g_{{ij}}(b)g_{{jk}}(b)=g_{{ik}}(b)$.
Readers familiar with Čech cohomology may recognize condition 3), it is often called the cocycle condition. Note, this imples that $g_{{ii}}(b)$ is the identity in $G$ for each $b$, and $g_{{ij}}(b)=g_{{ji}}(b)^{{1}}$.
If the total space $E$ is homeomorphic to the product $B\times F$ so that the bundle projection is essentially projection onto the first factor, then $\pi:E\to B$ is called a trivial bundle. Some examples of fiber bundles are vector bundles and covering spaces.
There is a notion of morphism of fiber bundles $E,E^{{\prime}}$ over the same base $B$ with the same structure group $G$. Such a morphism is a $G$equivariant map $\xi:E\to E^{{\prime}}$, making the following diagram commute
$\xymatrix{E\ar[rr]^{\xi}\ar[dr]_{\pi}&&E^{{\prime}}\ar[dl]^{{\pi^{{\prime}}}}% \\ &B&}.$ 
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