2: Locally Trivial Surjection of Rank k

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definition of locally trivial surjection of rank $k$

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of surjection.
• The reader knows a definition of neighborhood of point.
• The reader knows a definition of homeomorphism.
• The reader knows a definition of %field name% vectors space.
• The reader knows a definition of %category name% isomorphism.
• The reader knows a definition of Euclidean topological space.
• The reader knows a definition of product topological space.

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Target Context

• The reader will have a definition of locally trivial surjection of rank $k$.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$T$: $\in \{\text{ the topological spaces }\}$
$E$: $\in \{\text{ the topological spaces }\}$
$\ast \pi$: $:E\to T$, $\in \{\text{ the surjections }\}\cap \{\text{ the continuous maps }\}$
$k$: $\in \mathbb{N}\setminus \{0\}$
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Conditions:
$\mathrm{\forall }t\in T({\pi }^{-1}(p)\in \{\text{ the }k\text{ -dimensional }\mathbb{R}\text{ vectors spaces }\})$
$\wedge$
$\mathrm{\forall }t\in T(\mathrm{\exists }{U}_t\in \{\text{ the open neighborhoods of }t\text{ on }T\},\mathrm{\exists }\mathrm{\Phi }:{\pi }^{-1}(U_t)\to U_t\times {\mathbb{R}}^k(\mathrm{\Phi }\in \{\text{ the homeomorphisms }\}\wedge \mathrm{\forall }{t}^{\prime }\in U_t(\mathrm{\Phi }{|}_{{\pi }^{-1}(t^{\prime })}:{\pi }^{-1}(t^{\prime })\to \{t^{\prime }\}\times {\mathbb{R}}^k\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\})))$
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${\mathbb{R}}^k$ is the Euclidean topological space; $U_t\times {\mathbb{R}}^k$ is the product topological space.

$\{t^{\prime }\}\times {\mathbb{R}}^k$ is the vectors space canonically 'vectors spaces - linear morphisms' isomorphic to ${\mathbb{R}}^k$.

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2: Note

$U_t$ is called "trivializing open subset".

$\mathrm{\Phi }$ is called "local trivialization".

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References

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