definition of product topology
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description 1
- 2: Natural Language Description 1
- 3: Structured Description 2
- 4: Natural Language Description 2
- 5: Note
Starting Context
- The reader knows a definition of product set.
- The reader knows a definition of topological space.
Target Context
- The reader will have a definition of product topology.
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 1
Here is the rules of Structured Description.
Entities:
\( A\): \(\in \{\text{ the possibly uncountable infinite index sets }\}\)
\( \{T_\alpha \vert \alpha \in A\}\): \(T_\alpha \in \{\text{ the topological spaces }\}\)
\( T\): \(= \times_{\alpha \in A} T_\alpha\)
\( B\): \(= \{\times_{\alpha \in A} U_\alpha \subseteq T \vert U_\alpha \in \{\text{ the open subsets of } T_\alpha\} \text{ where only finite of } U_\alpha \text{ s are not } T_\alpha s\}\)
\(*O\): \(\in \{\text{ the topologies of } T\}\)
//
Conditions:
\(O = \) the topology generated by the basis, \(B\)
//
Each element of \(B\) is called 'basic open set on \(T\)'.
2: Natural Language Description 1
For any possibly uncountable infinite index set, \(A\), any topological spaces, \(\{T_\alpha \vert \alpha \in A\}\), the product set, \(T := \times_{\alpha \in A} T_\alpha\), and \(B := \{\times_{\alpha \in A} U_\alpha \subseteq T \vert U_\alpha \in \{\text{ the open subsets of } T_\alpha\} \text{ where only finite of } U_\alpha \text{ s are not } T_\alpha s\}\), the topology, \(O\), generated by the basis, \(B\), where any element of \(B\) is called 'basic open set on \(T\)'
3: Structured Description 2
Here is the rules of Structured Description.
Entities:
\( \{T_1, ..., T_n\}\): \(T_j \in \{\text{ the topological spaces }\}\)
\( T\): \(= T_1 \times ... \times T_n\)
\( B\): \(= \{U_1 \times ... \times U_n \subseteq T \vert U_j \in \{\text{ the open subsets of } T_j\}\}\)
\(*O\): \(\in \{\text{ the topologies of } T\}\)
//
Conditions:
\(O = \) the topology generated by the basis, \(B\)
//
Each element of \(B\) is called 'basic open set on \(T\)'.
4: Natural Language Description 2
For any topological spaces, \(\{T_1, ..., T_n\}\), the product set, \(T := T_1 \times ... \times T_n\), and \(B := \{U_1 \times ... \times U_n \subseteq T \vert U_j \in \{\text{ the open subsets of } T_j\}\}\), the topology, \(O\), generated by the basis, \(B\), where any element of \(B\) is called 'basic open set on \(T\)'
5: Note
The definition is valid because \(B\) satisfies Description 2 of the criteria for any collection of open sets to be a basis. In fact, 1) is satisfied because \(B\) contains \(\times_{\alpha \in A} T_\alpha\) or \(T_1 \times T_2 \times . . . \times T_n\); 2) is satisfied because for any \(B_1 = \times_{\alpha \in A} U_\alpha\) or \(B_1 = U_1 \times U_2 \times . . . \times U_n\) and \(B_2 = \times_{\alpha \in A} V_\alpha\) or \(B_2 = V_1 \times V_2 \times . . . \times V_n\) such that \(B_1 \cap B_2 \neq \emptyset\) and any point, \(p \in B_1 \cap B_2\), \(p (\alpha) \in U_\alpha \cap V_\alpha\) for each \(\alpha\) or \(p^i \in U_i \cap V_i\) for each \(i\) where \(p = (p^1, p^2, . . ., p^n)\), but as \(U_\alpha \cap V_\alpha\) or \(U_i \cap V_i\) is open on \(T_\alpha\) or \(T_i\), there is an open set, \(p (\alpha) \in W_\alpha \subseteq U_\alpha \cap V_\alpha\) where only finite of \(W_\alpha\) s are not \(T_\alpha\) or \(p^i \in W_i \subseteq U_i \cap V_i\), and \(p \in B_3 = \times_{\alpha \in A} W_\alpha\) or \(p \in B_3 = W_1 \times W_2 \times . . . \times W_n\) where \(B_3 \subseteq B_1 \cap B_2\) and \(B_3 \in B\).
By Description 1 of the criteria for any collection of open sets to be a basis, any subset, \(S \subseteq T\), is open on \(T\) if and only if \(S = \cup_{\gamma \in C} \times_{\alpha \in A} U_{\alpha, \gamma}\) or \(S = \cup_{\gamma \in C} U_{1, \gamma} \times U_{2, \gamma} \times, . . ., \times U_{n, \gamma}\) for a possibly uncountable index set, \(C\), where \(U_{\alpha, \gamma} \subseteq T_\alpha\) or \(U_{i, \gamma} \subseteq T_i\) is open on \(T_\alpha\) or \(T_i\) and only finite of \(U_{\alpha, \gamma}\) s are not \(T_\alpha\) for each \(\gamma\).
