237: Product Topology

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definition of product topology

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Topics

About: topological space

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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

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Starting Context

• The reader knows a definition of product set.
• The reader knows a definition of topological space.

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Target Context

• The reader will have a definition of product topology.

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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.

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Main Body

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1: Structured Description 1

Here is the rules of Structured Description.

Entities:
$A$: $\in \{\text{ the possibly uncountable infinite index sets }\}$
$\{T_{\alpha }|\alpha \in A\}$: $T_{\alpha }\in \{\text{ the topological spaces }\}$
$T$: $={\times }_{\alpha \in A}{T}_{\alpha }$
$B$: $=\{{\times }_{\alpha \in A}{U}_{\alpha }\subseteq T|U_{\alpha }\in \{\text{ the open subsets of }{T}_{\alpha }\}\text{ where only finite of }{U}_{\alpha }\text{ s are not }{T}_{\alpha }s\}$
$\ast O$: $\in \{\text{ the topologies of }T\}$
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Conditions:
$O=$ the topology generated by the basis, $B$
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Each element of $B$ is called 'basic open set on $T$'.

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2: Natural Language Description 1

For any possibly uncountable infinite index set, $A$, any topological spaces, $\{T_{\alpha }|\alpha \in A\}$, the product set, $T:={\times }_{\alpha \in A}{T}_{\alpha }$, and $B:=\{{\times }_{\alpha \in A}{U}_{\alpha }\subseteq T|U_{\alpha }\in \{\text{ the open subsets of }{T}_{\alpha }\}\text{ where only finite of }{U}_{\alpha }\text{ s are not }{T}_{\alpha }s\}$, the topology, $O$, generated by the basis, $B$, where any element of $B$ is called 'basic open set on $T$'

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3: Structured Description 2

Here is the rules of Structured Description.

Entities:
$\{T_1,...,T_n\}$: $T_j\in \{\text{ the topological spaces }\}$
$T$: $=T_1\times ...\times T_n$
$B$: $=\{U_1\times ...\times U_n\subseteq T|U_j\in \{\text{ the open subsets of }{T}_j\}\}$
$\ast O$: $\in \{\text{ the topologies of }T\}$
//

Conditions:
$O=$ the topology generated by the basis, $B$
//

Each element of $B$ is called 'basic open set on $T$'.

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4: Natural Language Description 2

For any topological spaces, $\{T_1,...,T_n\}$, the product set, $T:=T_1\times ...\times T_n$, and $B:=\{U_1\times ...\times U_n\subseteq T|U_j\in \{\text{ the open subsets of }{T}_j\}\}$, the topology, $O$, generated by the basis, $B$, where any element of $B$ is called 'basic open set on $T$'

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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 $B_1\cap B_2\ne \mathrm{\varnothing }$ 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 $\gamma$.

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References

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