definition of trivializing open subset and local trivialization
Topics
About: topological space
The table of contents of this article
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.
Target Context
- The reader will have a definition of trivializing open subset and local trivialization.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( E\): \(\in \{\text{ the topological spaces }\}\)
\( \pi\): \(: E \to T\), \(\in \{\text{ the continuous maps }\}\)
\( k\): \(\in \mathbb{N} \setminus \{0\}\)
\(*U\): \(\in \{\text{ the open subsets of } T\}\)
\(*\Phi\): \(\pi^{-1} (U) \to U \times \mathbb{R}^k\)
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Conditions:
\(\Phi \in \{\text{ the homeomorphisms }\} \land \forall t' \in U (\Phi \vert_{\pi^{-1} (t')}: \pi^{-1} (t') \to \{t'\} \times \mathbb{R}^k \in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\})\)
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\(\mathbb{R}^k\) is the Euclidean topological space; \(U_t \times \mathbb{R}^k\) is the product topological space.
\(\{t'\} \times \mathbb{R}^k\) is the vectors space canonically 'vectors spaces - linear morphisms' isomorphic to \(\mathbb{R}^k\).
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.
