definition of canonical topology for finite-dimensional real vectors space
Topics
About: vectors space
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of topology.
- The reader admits the proposition that for any finite dimensional real vectors space, the topology defined by the Euclidean topology of the coordinates space based on any basis does not depend on the choice of basis.
Target Context
- The reader will have a definition of canonical topology for finite-dimensional real vectors space.
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:
\( V\): \(\in \{\text{ the }d\text{ -dimensional real vectors spaces }\}\)
\( \{b_1, ..., b_d\}\): \(\subseteq V\), \(\in \{\text{ the bases of } V\}\)
\( \mathbb{R}^d\): \(= \text{ the Euclidean topological space }\)
\( f\): \(: V \to \mathbb{R}^d\), \(v \mapsto (v^1, ..., v^d)\) such that \(v = v^j b_j\)
\(*O\): \(= \{U \subseteq V \vert f (U) \in \{\text{ the topology of } \mathbb{R}^d\}\}\)
//
Conditions:
//
\(O\) does not depend on the choice of \(\{b_1, ..., b_d\}\), by the proposition that for any finite dimensional real vectors space, the topology defined by the Euclidean topology of the coordinates space based on any basis does not depend on the choice of basis.
2: Natural Language Description
For any \(d\)-dimensional real vectors space, \(V\), any basis, \(\{b_1, ..., b_d\} \subseteq V\), the Euclidean topological space, \(\mathbb{R}^d\), and the map, \(f: V \to \mathbb{R}^d\), \(v \mapsto (v^1, ..., v^d)\) such that \(v = v^j b_j\), the topology, \(O = \{U \subseteq V \vert f (U) \in \{\text{ the topology of } \mathbb{R}^d\}\}\)
