826: Trivializing Open Subset and Local Trivialization

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definition of trivializing open subset and local trivialization

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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 continuous map.
• 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 trivializing open subset and local trivialization.

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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:
$T$: $\in \{\text{ the topological spaces }\}$
$E$: $\in \{\text{ the topological spaces }\}$
$\pi$: $:E\to T$, $\in \{\text{ the continuous maps }\}$
$k$: $\in \mathbb{N}\setminus \{0\}$
$\ast U$: $\in \{\text{ the open subsets of }T\}$
$\ast \mathrm{\Phi }$: ${\pi }^{-1}(U)\to U\times {\mathbb{R}}^k$
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Conditions:
$\mathrm{\Phi }\in \{\text{ the homeomorphisms }\}\wedge \mathrm{\forall }{t}^{\prime }\in U(\mathrm{\Phi }{|}_{{\pi }^{-1}(t^{\prime })}:{\pi }^{-1}(t^{\prime })\to \{t^{\prime }\}\times {\mathbb{R}}^k\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\})$
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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

Usually, 'trivializing open subset' and 'local trivialization' are talked about for a vectors bundle, but a trivializing open subset and a local trivialization are possible also for other cases.

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References

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