definition of locally trivial surjection of rank \(k\)
Topics
About: topological space
The table of contents of this article
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.
Target Context
- The reader will have a definition of locally trivial surjection of rank \(k\).
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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( E\): \(\in \{\text{ the topological spaces }\}\)
\(*\pi\): \(: E \to T\), \(\in \{\text{ the surjections }\} \cap \{\text{ the continuous maps }\}\)
\( k\): \(\in \mathbb{N} \setminus \{0\}\)
//
Conditions:
\(\forall t \in T (\pi^{-1} (p) \in \{\text{ the } k \text{ -dimensional } \mathbb{R} \text{ vectors spaces }\})\)
\(\land\)
\(\forall t \in T (\exists U_t \in \{\text{ the open neighborhoods of } t \text{ on } T\}, \exists \Phi: \pi^{-1} (U_t) \to U_t \times \mathbb{R}^k (\Phi \in \{\text{ the homeomorphisms }\} \land \forall t' \in U_t (\Phi \vert_{\pi^{-1} (t')}: \pi^{-1} (t') \to \{t'\} \times \mathbb{R}^k \in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\})))\)
//
\(\mathbb{R}^k\) 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
\(U_t\) is called "trivializing open subset".
\(\Phi\) is called "local trivialization".
