definition of maps homotopic relative to subset of domain
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
Starting Context
- The reader knows a definition of continuous map.
- The reader knows a definition of product topology.
Target Context
- The reader will have a definition of maps homotopic relative to subset of domain.
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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\( S\): \(\subseteq T_1\)
\(*f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(*f'\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
//
Conditions:
\(\exists F: T_1 \times I \to T_2, \in \{\text{ the continuous maps }\}\), where \(I\) is \([0, 1] \subseteq \mathbb{R}\)
(
\(\forall p \in T_1\)
(
\(F (p, 0) = f (p)\)
\(\land\)
\(F (p, 1) = f' (p)\)
)
\(\land\)
\(\forall p \in S, \forall s \in I\)
(
\(F (p, s) = f (p) = f' (p)\)
)
).
//
\(F\) is called "relative homotopy".
\(f \simeq f' rel S\) denotes the relation.
2: Natural Language Description
For any topological spaces, \(T_1, T_2\), and any subset, \(S \subseteq T_1\), any continuous maps, \(f, f': T_1 \to T_2\), such that there is a continuous map (called "relative homotopy"), \(F: T_1 \times I \to T_2\), where \(I\) is \([0, 1] \subseteq \mathbb{R}\), such that for each \(p \in T_1\), \(F (p, 0) = f (p)\) and \(F (p, 1) = f' (p)\), and for each \(p \in S\) and each \(s \in I\), \(F (p, s) = f (p) = f' (p)\), denoted as \(f \simeq f' rel S\)
