description/proof of that for \(2\) continuous maps from same domain into same codomain and finite disjoint closed cover of domain, if restrictions of maps on each element of cover are homotopic relative to subset, maps are homotopic relative to union of subsets
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of maps homotopic relative to subset of domain.
- The reader admits the proposition that for any product topological space, the product of any closed subsets is closed.
- The reader admits the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous.
- The reader admits the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
Target Context
- The reader will have a description and a proof of the proposition that for any \(2\) continuous maps from any same domain into any same codomain and any finite disjoint closed cover of the domain, if the restrictions of the maps on each element of the cover are homotopic relative to any subset, the maps are homotopic relative to the union of the subsets.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(\in T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(f'\): \(\in T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(\{C_j \in \{\text{ the closed subsets of } T_1\} \vert j \in J\}\): where \(J \in \{\text{ the finite index sets }\}\), \(\forall j, j' \in J \text{ such that } j \neq j' (C_j \cap C_{j'} = \emptyset)\), and \(\cup_{j \in J} C_j = T_1\)
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Statements:
\(\forall j \in J (f \vert_{C_j} \simeq f' \vert_{C_j} rel S_j)\)
\(\implies\)
\(f \simeq f' rel \cup_{j \in J} S_j\)
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2: Note
Each \(f \vert_{C_j}: C_j \to T_2\) or \(f' \vert_{C_j}: C_j \to T_2\) is inevitably continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous, so, talking about being homotopic of \(f \vert_{C_j}\) and \(f' \vert_{C_j}\) makes sense.
\(\{C_j \in \{\text{ the closed subsets of } T_1\} \vert j \in J\}\) needs to be disjoint for this proposition, because otherwise, when the homotopy between \(f\) and \(f'\) was constructed from the homotopies between the restricted maps, the consistency would be a concern.
A typical case that this proposition applies is that \(T_1\) is a topological sum, \(T_{1, 1} + ... + T_{1, n}\): \(\{T_{1, j}\}\) is a finite disjoint closed cover of \(T_1\).
3: Proof
Whole Strategy: Step 1: for each \(j \in J\), take a homotopy, \(F_j: C_j \times I \to T_2\) between \(f \vert_{C_j}\) and \(f' \vert_{C_j}\) relative to \(S_j\); Step 2: take \(F: T_1 \times I \to T_2\) such that \(F \vert_{C_j \times I} = F_j\), and see that \(F\) is a homotopy between \(f\) and \(f'\) relative to \(\cup_{j \in J} S_j\).
Step 1:
Let \(j \in J\) be any.
As \(f \vert_{C_j} \simeq f' \vert_{C_j} rel S_j\), there is a homotopy, \(F_j: C_j \times I \to T_2\), such that \(F_j (c_j, 0) = f (c_j)\), \(F_j (c_j, 1) = f' (c_j)\), and for each \(s_j \in S_j\), \(F_j (s_j, j) = f (s_j) = f' (s_j)\).
Step 2:
Let us take \(F: T_1 \times I \to T_2\) such that \(F \vert_{C_j \times I} = F_j\).
\(F\) is well-defined, because \(\{C_j \times I \vert j \in J\}\) is a disjoint cover of \(T_1 \times I\).
Each \(C_j \times I\) is a closed subset of \(T_1 \times I\), by the proposition that for any product topological space, the product of any closed subsets is closed.
So, \(\{C_j \times I \vert j \in J\}\) is a closed cover of \(T_1 \times I\).
\(F\) is continuous, by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous: the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
For each \(t_1 \in T_1\), \(t_1 \in C_j\) for a \(j \in J\), and \(F (t_1, 0) = F_j (t_1, 0) = f (t_1)\).
For each \(t_1 \in T_1\), \(t_1 \in C_j\) for a \(j \in J\), and \(F (t_1, 1) = F_j (t_1, 1) = f' (t_1)\).
For each \(s \in \cup_{j \in J} S_j\), \(s \in S_j \subseteq C_j\) for a \(j \in J\), and for each \(r \in I\), \(F (s, r) = F_j (s, r) = f (s) = f' (s)\).
So, \(F\) is a homotopy between \(f\) and \(f'\) relative to \(\cup_{j \in J} S_j\).
So, \(f \simeq f' rel \cup_{j \in J} S_j\).
