description/proof of for topological space and set of continuous maps from space into Euclidean topological space s.t. set of preimages of nonzero is locally finite, sum of maps is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean topological space.
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of map preimage of subset of codomain.
- The reader knows a definition of locally finite set of subsets of topological space.
- The reader admits the proposition that for any topological space and any set of maps from the space into any ring or module such that the set of the preimages of nonzero is locally finite, the support of the sum of the maps is contained in the union of the supports of the maps.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
- The reader admits the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space and any set of continuous maps from the space into any Euclidean topological space such that the set of the preimages of nonzero is locally finite, the sum of the maps is continuous.
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:
\(T\): \(= \{\text{ the topological spaces }\}\)
\(\mathbb{R}^d\): \(\in \{\text{ the Euclidean topological spaces }\}\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{f_j: T \to \mathbb{R}^d, \in \{\text{ the continuous maps }\} \vert j \in J\}\): such that \(\{{f_j}^{-1} (\mathbb{R}^d \setminus \{0\}) \vert j \in J\}\) is locally finite
\(\sum_{j \in J} f_j\): \(: T \to \mathbb{R}^d, t \mapsto \sum_{j \in J^`_t} f_j (t) \text{ for any } J^`_t \in \{\text{ the finite subsets of } J\} \text{ such that } \forall j \in J \setminus J^`_t (f_j (t) = 0)\)
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Statements:
\(\sum_{j \in J} f_j \in \{\text{ the continuous maps }\}\)
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2: Note
\(\sum_{j \in J} f_j\) is valid, by Note for the proposition that for any topological space and any set of maps from the space into any ring or module such that the set of the preimages of nonzero is locally finite, the support of the sum of the maps is contained in the union of the supports of the maps.
When \(\{Supp (f_j) \vert j \in J\}\) is locally finite, \(\{{f_j}^{-1} (\mathbb{R}^d \setminus \{0\}) \vert j \in J\}\) is locally finite, because \({f_j}^{-1} (\mathbb{R}^d \setminus \{0\}) \subseteq Supp (f_j)\).
3: Proof
Whole Strategy: Step 1: for each \(t \in T\), take an open neighborhood of \(t\), \(U_t\), that intersects only some finite \({f_j}^{-1} (\mathbb{R}^d \setminus \{0\})\) s, see that each \((\sum_{j \in J} f_j) \vert_{U_t}\) is continuous, and apply the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.
Step 1:
For each \(t \in T\), there is a neighborhood of \(t\), \(N_t \subseteq T\), such that there is a finite \(J^`_t \subseteq J\) such that for each \(j \in J \setminus J^`_t\), \(N_t \cap {f_j}^{-1} (\mathbb{R}^d \setminus \{0\}) = \emptyset\), then, there is an open neighborhood of \(t\), \(U_t \subseteq T\), such that \(U_t \subseteq N_t\), and \(U_t \cap {f_j}^{-1} (\mathbb{R}^d \setminus \{0\}) = \emptyset\) for each \(j \in J \setminus J^`_t\).
\((\sum_{j \in J} f_j) \vert_{U_t} = \sum_{j \in J^`_t} f_j \vert_{U_t}\), because for each \(j \in J\) such that \(j \notin J^`_t\), \(f_j \vert_{U_t} = 0\), because \(U_t \cap {f_j}^{-1} (\mathbb{R}^d \setminus \{0\}) = \emptyset\), which means that for each \(t' \in U_t\), \(t' \notin {f_j}^{-1} (\mathbb{R}^d \setminus \{0\})\), which means that \(f_j (t') = 0\).
So, \((\sum_{j \in J} f_j) \vert_{U_t}\) is continuous, as a finite sum of some continuous maps into \(\mathbb{R}^d\): each \(f_j \vert_{U_t}\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
\(\{U_t \vert t \in T\}\) is an open cover of \(T\).
So, \(\sum_{j \in J} f_j\) is continuous, by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.
