description/proof of that continuous map from contractible topological space into topological space is homotopic to constant map
Topics
About: topological space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that any continuous map from any contractible topological space into any topological space is homotopic to a constant 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:
\(T_1\): \(\in \{\text{ the contractible topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
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Statements:
\(\exists c_{p_2}: T_1 \to T_2, \in \{\text{ the constant maps }\} (f \simeq c_{p_2})\)
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2: Proof
Whole Strategy: Step 1: take a constant map, \(c_{p_1}: T_1 \to T_1, t \mapsto p_1\), and a homotopy from \(id_{T_1}\) to \(c_{p_1}\), \(g: T_1 \times I \to T_1\); Step 2: take \(f \circ g\) and see that \(f \circ g\) is a homotopy from \(f\) to the constant map to \(p_2 := f (p_1)\).
Step 1:
There are a constant map, \(c_{p_1}: T_1 \to T_1, t \mapsto p_1\), and a homotopy from \(id_{T_1}: T_1 \to T_1\) to \(c_{p_1}\), \(g: T_1 \times I \to T_1\), by the definition of contractible topological space.
Step 2:
Let us take \(f \circ g: T_1 \times I \to T_2\).
\(f \circ g\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
For each \(t \in T_1\), \(f \circ g (t, 0) = f \circ id_{T_1} (t) = f (t)\) and \(f \circ g (t, 1) = f \circ c_{p_1} (t) = f (p_1) := p_2\), which is a constant map to \(p_2\), \(:= c_{p_2}\).
So, \(f \simeq c_{p_2}\).
