Category:Definitions/Fiber Bundles

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This category contains definitions related to Fiber Bundles.
Related results can be found in Category:Fiber Bundles.

Let $M, E, F$ be topological spaces.

Let $\pi: E \to M$ be a continuous surjection.

Let $\mathcal U := \left\{ {U_\alpha \subseteq M: \alpha \in I} \right\}$ be an open cover of $M$ with index set $I$.

Let $\operatorname{pr}_{1, \alpha} : U_\alpha \times F \to U_\alpha$ be the first projection on $U_\alpha \times F$.

Let there exist homeomorphisms:

$\chi_\alpha : \pi^{-1} \left({U_\alpha}\right) \to U_\alpha \times F$

such that for all $\alpha \in I$:

$\pi {\restriction}_{U_\alpha} = \operatorname{pr}_{1, \alpha} \mathop \circ \chi_\alpha$

where $\pi {\restriction}_{U_\alpha}$ is the restriction of $\pi$ to $U_\alpha \in \mathcal U$.

Then the ordered tuple $\left({E, M, \pi, F}\right)$ is called a fiber bundle over $M$.