description/proof of that \(2\) continuous maps from contractible topological space into path-connected topological space are homotopic
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of contractible topological space.
- The reader knows a definition of path-connected topological space.
- The reader knows a definition of homotopic maps.
- The reader admits the proposition that any continuous map from any contractible topological space into any topological space is homotopic to a constant map.
- The reader admits the proposition that any \(2\) constant maps from any topological space into any path-connected topological space are homotopic.
- The reader admits the proposition that on the set of the continuous maps between any topological spaces, being homotopic is an equivalence relation.
Target Context
- The reader will have a description and a proof of the proposition that any \(2\) continuous maps from any contractible topological space into any path-connected topological space are homotopic.
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:
\(T_1\): \(\in \{\text{ the contractible topological spaces }\}\)
\(T_2\): \(\in \{\text{ the path-connected topological spaces }\}\)
\(f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(f'\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
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Statements:
\(f \simeq f'\), where \(\simeq\) means being homotopic
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2: Proof
Whole Strategy: Step 1: take a constant map, \(c_{p_2}\), that is homotopic to \(f\) and a constant map, \(c_{p'_2}\), that is homotopic to \(f'\); Step 2: see that \(c_{p_2}\) and \(c_{p'_2}\) are homotopic; Step 3: see that \(f\) and \(f'\) are homotopic.
Step 1:
There is a constant map, \(c_{p_2}: T_1 \to T_2, t \mapsto p_2\), that is homotopic to \(f\), by the proposition that any continuous map from any contractible topological space into any topological space is homotopic to a constant map, so, \(f \simeq c_{p_2}\).
There is a constant map, \(c_{p'_2}: T_1 \to T_2, t \mapsto p'_2\), that is homotopic to \(f'\), by the proposition that any continuous map from any contractible topological space into any topological space is homotopic to a constant map, so, \(f' \simeq c_{p'_2}\).
Step 2:
\(c_{p_2}\) and \(c_{p'_2}\) are homotopic, by the proposition that any \(2\) constant maps from any topological space into any path-connected topological space are homotopic, so, \(c_{p_2} \simeq c_{p'_2}\).
Step 3:
As \(\simeq\) is an equivalence relation, by the proposition that on the set of the continuous maps between any topological spaces, being homotopic is an equivalence relation, \(f \simeq c_{p_2} \simeq c_{p'_2} \simeq f'\) implies that \(f \simeq f'\).
