description/proof of that for topological space and set of maps from space into ring or module s.t. set of preimages of nonzero is locally finite, support of sum of maps is contained in union of supports of maps
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of support of map from topological space into ring or module.
- The reader knows a definition of locally finite set of subsets of topological space.
- The reader admits the proposition that for any locally finite set of subsets of any topological space, the closure of the union of the subsets is the union of the closures of the subsets.
Target Context
- The reader will have a description and a proof of 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.
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:
\(T\): \(= \{\text{ the topological spaces }\}\)
\(S\): \(\in \{\text{ the rings }\} \cup \{\text{ the modules }\}\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{f_j: T \to S \vert j \in J\}\): such that \(\{{f_j}^{-1} (S \setminus \{0\}) \vert j \in J\}\) is locally finite
\(\sum_{j \in J} f_j\): \(: T \to S, t \mapsto \sum_{j \in J^`_t} f_j (t) \text{ for any } J^`_t \in \{\text{ the finite subsets of } J\} \text{ such that } \forall j \in J \setminus J^`_t (f_j (t) = 0)\)
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Statements:
\(Supp (\sum_{j \in J} f_j) \subseteq \cup_{j \in J} Supp (f_j)\)
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2: Note
That \(\{{f_j}^{-1} (S \setminus \{0\}) \vert j \in J\}\) is locally finite implies that for each \(t \in T\), there is a finite \(J^`_t \subseteq J\) such that for each \(j \in J \setminus J^`_t\), \(f_j (t) = 0\), because there is a neighborhood of \(t\), \(N_t\), such that there is a finite \(J^`_t \subseteq J\) such that for each \(j \in J \setminus J^`_t\), \(N_t \cap {f_j}^{-1} (S \setminus \{0\}) = \emptyset\), then, \(f_j (t) = 0\), because otherwise, \(t \in N_t \cap {f_j}^{-1} (S \setminus \{0\})\), a contradiction.
And while there can be some multiple such \(J^`_t\) s, \(\sum_{j \in J^`_t} f_j (t)\) does not depend on the choice of \(J^`_t\), because any \(J^`_t\) needs to contain all the \(j\) s such that \(f_j (t) \neq 0\) and adding any other \(j\) s does not influence the result: it is natural to take the \(J^`\) such that for each \(j \in J^`\), \(f_j (t) \neq 0\).
So, the definition of \(\sum_{j \in J} f_j\) is valid.
The condition that \(\{Supp (f_j) \vert j \in J\}\) is locally finite is a sufficient condition for that \(\{{f_j}^{-1} (S \setminus \{0\}) \vert j \in J\}\) is locally finite, because as \({f_j}^{-1} (S \setminus \{0\}) \subseteq Supp (f_j)\), if \(N_t\) intersects only some finite \(Supp (f_j)\) s, \(N_t\) can intersect only the corresponding finite \({f_j}^{-1} (S \setminus \{0\})\) s: if \(N_t\) does not intersect a \(Supp (f_j)\), \(N_t\) does not intersect the corresponding \({f_j}^{-1} (S \setminus \{0\})\).
3: Proof
Whole Strategy: Step 1: see that \((\sum_{j \in J} f_j)^{-1} (S \setminus \{0\}) \subseteq \cup_{j \in J} {f_j}^{-1} (S \setminus \{0\})\); Step 2: see that \(\overline{\cup_{j \in J} {f_j}^{-1} (S \setminus \{0\})} = \cup_{j \in J} \overline{{f_j}^{-1} (S \setminus \{0\})}\).
Step 1:
\(Supp (\sum_{j \in J} f_j) = \overline{(\sum_{j \in J} f_j)^{-1} (S \setminus \{0\})}\), by the definition of support.
\(Supp (f_j) = \overline{{f_j}^{-1} (S \setminus \{0\})}\), by the definition of support.
Let us see that \((\sum_{j \in J} f_j)^{-1} (S \setminus \{0\}) \subseteq \cup_{j \in J} {f_j}^{-1} (S \setminus \{0\})\).
For each \(t \in (\sum_{j \in J} f_j)^{-1} (S \setminus \{0\})\), \((\sum_{j \in J} f_j) (t) \in S \setminus \{0\}\), which implies that \(f_j (t) \in S \setminus \{0\}\) for a \(j \in J\), so, \(t \in {f_j}^{-1} (S \setminus \{0\})\), so, \(t \in \cup_{j \in J} {f_j}^{-1} (S \setminus \{0\})\).
Step 2:
So, \(Supp (\sum_{j \in J} f_j) = \overline{(\sum_{j \in J} f_j)^{-1} (S \setminus \{0\})} \subseteq \overline{\cup_{j \in J} {f_j}^{-1} (S \setminus \{0\})} = \cup_{j \in J} \overline{{f_j}^{-1} (S \setminus \{0\})}\), by the proposition that for any locally finite set of subsets of any topological space, the closure of the union of the subsets is the union of the closures of the subsets, \(= \cup_{j \in J} Supp (f_j)\).
