A description/proof of that compositions of homotopic maps are homotopic
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 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.
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: Description
For any topological spaces, \(T_1, T_2, T_3\), any homotopic maps, \(f_1, f_2: T_1 \to T_2\), and any homotopic maps, \(f'_1, f'_2: T_2 \to T_3\), \(f'_1 \circ f_1\) and \(f'_2 \circ f_2\) are homotopic.
2: Proof
There are some homotopies, \(f_3: T_1 \times I \to T_2\) and \(f'_3: T_2 \times I \to T_3\) such that \(f_3 (p, 0) = f_1 (p)\), \(f_3 (p, 1) = f_2 (p)\), \(f'_3 (p, 0) = f'_1 (p)\), and \(f'_3 (p, 1) = f'_2 (p)\).
Let us define \(f'': T_1 \times I \to T_2 \times I \to T_3\), \((p, i) \mapsto (f_3 (p, i), i) \mapsto f'_3 (f_3 (p, i), i)\). \(f''\) is continuous as a composition of continuous maps, while the 1st half of the composition is continuous by the proposition that any map from any topological space into any product topological space is continuous if and only if the composition of the projection of the map into each element of the product after the map is continuous. \(f'' (p, 0) = f'_3 (f_3 (p, 0), 0) = f'_3 (f_1 (p), 0) = f'_1 (f_1 (p))\). \(f'' (p, 1) = f'_3 (f_3 (p, 1), 1) = f'_3 (f_2 (p), 1) = f'_2 (f_2 (p))\).
