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2010-12-25T00:53:04Z

Created page with '==Definition== Suppose <math>E,B,F</math> are topological spaces. A continuous map <math>p:E \to B</math> is termed a '''fiber bundle''' with fiber <math>F</math> if: *...'

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==Definition==

Suppose <math>E,B,F</math> are [[topological space]]s. A [[continuous map]] <math>p:E \to B</math> is termed a '''fiber bundle''' with fiber <math>F</math> if:

* <math>p</math> is surjective
* For any point <math>b \in B</math>, the fiber <math>p^{-1}(\{ b \})</math>, given the [[subspace topology]] from <math>E</math>, is homeomorphic to <math>F</math>.
* For any point <math>b \in B</math>, there exists an open subset <math>U</math> of <math>B</math> such that <math>b \in U</math>, and such that there exists a homeomorphism <math>\varphi: U \times F \to p^{-1}(U)</math> (where <math>U \times F</math> is endowed with the [[product topology]]) with the property that <math>p \circ \varphi</math> coincides with the projection map from <math>U \times F</math> to <math>U</math>. In other words, the bundle is ''locally trivial'', i.e., ''locally like a product space''.

We use the following terminology:

* The space <math>F</math> is termed the ''fiber space'' or ''fiber'' or ''fiber type''.
* The space <math>E</math> is termed the ''total space''.
* The space <math>B</math> is termed the ''base space''.

The letters <math>F,E,B</math> are typical in this context.

==Particular cases==

===Product bundle===

A special case of a fiber bundle is the ''trivial'' fiber bundle, where <math>E = B \times F</math> with <math>p:E \to B</math> being the projection onto the first coordinate. We can also think of this as being ''globally trivial.''

===Covering map===

When <math>F</math> is a [[discrete space]], then a fiber bundle with fiber <math>F</math> is a [[covering map]]. In fact, a covering map can be defined as a fiber bundle with discrete fiber.

Fiber bundle - Revision history

Vipul: Created page with '==Definition== Suppose E,B,F are topological spaces. A continuous map p:E \to B is termed a '''fiber bundle''' with fiber F if: *...'

Vipul: Created page with '==Definition== Suppose $E,B,F$ are topological spaces. A continuous map $p:E \to B$ is termed a '''fiber bundle''' with fiber $F$ if: *...'