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fibre (Definition)

Given a function $ f\colon X \longrightarrow Y$, a fibre is an inverse image of an element of $ Y$. That is given $ y \in Y$, $ f^{-1}(\{y\}) = \{ x \in X \mid f(x) = y \}$ is a fibre.

Example: Define $ f\colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ by $ f(x,y) = x^2 + y^2$. Then the fibres of $ f$ consist of concentric circles about the origin, the origin itself, and empty sets depending on whether we look at the inverse image of a positive number, zero, or a negative number respectively.

Example: Suppose $ M$ is a manifold, and $ \pi\colon TM\to M$ is the canonical projection from the tangent bundle $ TM$ to $ M$. Then fibres of $ \pi$ are the tangent spaces $ T_x(M)$ for $ x\in M$.



"fibre" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: level set

Other names:  fiber
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Cross-references: tangent spaces, tangent bundle, canonical projection, manifold, negative number, positive, empty sets, origin, concentric circles, inverse image, function
There are 42 references to this entry.

This is version 5 of fibre, born on 2002-08-07, modified 2005-06-10.
Object id is 3276, canonical name is Fibre.
Accessed 6645 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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