definition of section of continuous surjection
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of continuous map.
- The reader knows a definition of surjection.
Target Context
- The reader will have a definition of section of continuous surjection.
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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\( \pi\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous surjections }\}\)
\(*s\): \(: T_2 \to T_1\), \(\in \{\text{ the continuous maps }\}\)
//
Conditions:
\(\pi \circ s: T_2 \to T_2 = id\)
//
\(s\) is called "section of \(\pi\)".
2: Natural Language Description
For any topological spaces, \(T_1, T_2\), and any continuous surjection, \(\pi: T_1 \to T_2\), any continuous map, \(s: T_2 \to T_1\), such that \(\pi \circ s: T_2 \to T_2\) is the identity map, \(id\), is a section of \(\pi\)
3: Note
\(\pi\) needs to be surjective, because otherwise, there would be a \(t \in T_2\) that would not be mapped under \(\pi\), and then, \(\pi \circ s (t) = t\) would be impossible whatever \(s\) we chose, which means that \(\pi \circ s = id\) would be impossible.
