description/proof of that for contractible topological space, each constant topological space endomorphism is homotopic to identity map
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 homotopic maps.
- The reader admits the proposition that any contractible topological space is path-connected.
- 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 for any contractible topological space, each constant topological space endomorphism is homotopic to the identity map.
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:
\(T\): \(\in \{\text{ the contractible topological spaces }\}\)
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Statements:
\(\forall f: T \to T \in \{\text{ the constant maps }\} (f \simeq id_T)\)
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2: Note
While the definition of contractible topological space requires that there is a constant map homotopic to \(id_T\), this proposition claims that each constant map is homotopic to \(id_T\).
3: Proof
Whole Strategy: Step 1: see that \(T\) is path-connected; Step 2: take a constant map, \(c_{t'}\), such that \(id_T \simeq c_{t'}\), and see that for each constant map, \(c_{t''}\), \(c_{t'} \simeq c_{t''}\).
Step 1:
\(T\) is path-connected, by the proposition that any contractible topological space is path-connected.
Step 2:
There is a constant map, \(c_{t'}: T \to T, t \mapsto t'\), such that \(id_T \simeq c_{t'}\), by the definition of contractible topological space.
Let \(c_{t''}: T \to T, t \mapsto t''\) be any continuous map.
\(c_{t'} \simeq c_{t''}\), by the proposition that any \(2\) constant maps from any topological space into any path-connected topological space are homotopic.
\(id_T \simeq c_{t'} \simeq c_{t''}\) implies that \(id_T \simeq c_{t''}\), by the proposition that on the set of the continuous maps between any topological spaces, being homotopic is an equivalence relation.
