description/proof of that for product topological space and neighborhood of point, there is open neighborhood of point contained in neighborhood as product of some open neighborhoods of components of point
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of product topological space.
- The reader knows a definition of neighborhood of point on topological space.
Target Context
- The reader will have a description and a proof of the proposition that for any product topological space and any neighborhood of any point, there is an open neighborhood of the point contained in the neighborhood as the product of some open neighborhoods of the components of the point.
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:
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{T_j \in \{\text{ the topological spaces }\} \vert j \in J\}\):
\(\times_{j \in J} T_j\): \(= \text{ the product topological space }\)
\(t\): \(\in \times_{j \in J} T_j\)
\(N_t\): \(\in \{\text{ the neighborhoods of } t \text{ on } \times_{j \in J} T_j\}\)
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Statements:
\(\exists J^` \subseteq J \in \{\text{ the finite sets }\}, \exists \{U_{t^j} \in \{\text{ the open neighborhoods of } t^j \text{ on } T_j\} \vert j \in J^`\} (\times_{j \in J} U_{t^j} \in \{\text{ the open neighborhoods of } t \text{ on } \times_{j \in J} T_j\} \text{ where } U_{t^j} := T_j \text{ for each } j \in J \setminus J^` \land \times_{j \in J} U_{t^j} \subseteq N_t)\)
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2: Proof
Whole Strategy: Step 1: take an open neighborhood of \(t\), \(U_t \subseteq N_t\), and see that \(U_t = \cup_{l \in L} \times_{j \in J} U_{l, j}\); Step 2: choose a \(\times_{j \in J} U_{l, j}\) such that \(t \in \times_{j \in J} U_{l, j}\).
Step 1:
There is an open neighborhood of \(t\), \(U_t \subseteq \times_{j \in J} T_j\), such that \(U_t \subseteq N_t\), by the definition of neighborhood of point.
\(U_t = \cup_{l \in L} \times_{j \in J} U_{l, j}\), where \(L\) is a possibly uncountable index set and \(U_{l, j} \subseteq T_j\) is open such that for each \(l \in L\), only finite of \(U_{l, j}\) s are not \(T_j\) s, by Note for the definition of product topology.
Step 2:
As \(t \in U_t\), \(t \in \times_{j \in J} U_{l, j}\) for an \(l \in L\).
That means that for each \(j \in J\), the \(j\)-component, \(t^j \in U_{l, j}\).
So, \(U_{l, j}\) is an open neighborhood of \(t^j\).
Let us define for each \(j \in J\), \(U_{t^j} := U_{l, j}\).
There is the \(J^` \subseteq J\) such that for each \(j \in J^`\), \(U_{t^j} \subset T_j\), and for each \(j \in J \setminus J^`\), \(U_{t^j} = T_j\): \(J^`\) may be empty.
\(J^`\) is finite.
\(t \in \times_{j \in J} U_{t^j}\).
\(\times_{j \in j} U_{t^j} \subseteq \times_{j \in J} T_j\) is open: only finite of \(U_{t^j}\) s are not \(T_j\) s.
So, \(U_{t^j}\) is an open neighborhood of \(t\).
\(\times_{j \in j} U_{t^j} = \times_{j \in j} U_{l, j} \subseteq \cup_{l \in L} \times_{j \in J} U_{l, j} \subseteq U_t \subseteq N_t\).
