description/proof of that compositions of homotopic maps are homotopic with homotopy as this
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of homotopic maps.
- The reader admits the proposition that any map from any topological space into any product topological space is continuous if and only if each component map is continuous.
- The reader admits 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.
Target Context
- The reader will have a description and a proof of the proposition that for any homotopic maps from any 1st topological space into any 2nd topological space and any homotopic maps from the 2nd topological space into any 3rd topological space, the compositions of the homotopic maps are homotopic with a homotopy as this.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(T_3\): \(\in \{\text{ the topological spaces }\}\)
\(f_1\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(f'_1\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(f_2\): \(: T_2 \to T_3\), \(\in \{\text{ the continuous maps }\}\)
\(f'_2\): \(: T_2 \to T_3\), \(\in \{\text{ the continuous maps }\}\)
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Statements:
\(f_1 \simeq f'_1 \text{ with a homotopy } F_1 \land f_2 \simeq f'_2 \text{ with a homotopy } F_2\)
\(\implies\)
\(f_2 \circ f_1 \simeq f'_2 \circ f'_1\) with a homotopy, \(F: T_1 \times I \to T_3, (t_1, j) \mapsto F_2 (F_1 (t_1, j), j)\)
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2: Proof
Whole Strategy: Step 1: take any homotopy from \(f_1\) to \(f'_1\), \(F_1\) and any homotopy from \(f_2\) to \(f'_2\), \(F_2\); Step 2: take \(F: T_1 \times I \to T_3, (t_1, j) \mapsto F_2 (F_1 (t_1, j), j)\) and see that \(F\) is a homotopy from \(f_2 \circ f_1\) to \(f'_2 \circ f'_1\).
Step 1:
Let us take any homotopy from \(f_1\) to \(f'_1\), \(F_1: T_1 \times I \to T_2\): \(F_1 (t_1, 0) = f_1 (t_1)\) and \(F_1 (t_1, 1) = f'_1 (t_1)\).
Let us take any homotopy from \(f_2\) to \(f'_2\), \(F_2: T_2 \times I \to T_3\): \(F_2 (t_2, 0) = f_2 (t_2)\) and \(F_2 (t_2, 1) = f'_2 (t_2)\).
Step 2:
Let us take \(F: T_1 \times I \to T_3, (t_1, j) \mapsto F_2 (F_1 (t_1, j), j)\).
\(F': T_1 \times I \to T_2 \times I, (t_1, j) \mapsto (F_1 (t_1, j), j)\) is continuous, by the proposition that any map from any topological space into any product topological space is continuous if and only if each component map is continuous: \(: T_1 \times I \to I, (t_1, j) \mapsto j\) is continuous, because for each open neighborhood of \(j\), \(U_j \subseteq I\), \(T_1 \times U_j\) is mapped into \(U_j\) where \(T_1 \times U_j \subseteq T_1 \times I\) is an open neighborhood of \((t, j)\).
\(F = F_2 \circ F'\) 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 \(t_1 \in T_1\), \(F (t_1, 0) = F_2 (F_1 (t_1, 0), 0) = F_2 (f_1 (t_1), 0) = f_2 (f_1 (t_1)) = f_2 \circ f_1 (t_1)\) and \(F (t_1, 1) = F_2 (F_1 (t_1, 1), 1) = F_2 (f'_1 (t_1), 1) = f'_2 (f'_1 (t_1)) = f'_2 \circ f'_1 (t_1)\).
So, \(F\) is a homotopy from \(f_2 \circ f_1\) to \(f'_2 \circ f'_1\) and \(f_2 \circ f_1 \simeq f'_2 \circ f'_1\).
