525: Product Topological Space

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definition of product topological space

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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

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Starting Context

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

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Target Context

• The reader will have a definition of product topological space.

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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:
$A$: $\in \{\text{ the possibly uncountable infinite index sets }\}$
$\{T_{\alpha }|\alpha \in A\}$: $T_{\alpha }\in \{\text{ the topological spaces }\}$
$\ast T$: $={\times }_{\alpha \in A}{T}_{\alpha }$ with the product topology
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Conditions:
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2: Natural Language Description 1

For any possibly uncountable infinite index set, $A$, and any topological spaces, $\{T_{\alpha }|\alpha \in A\}$, the product set, $T:={\times }_{\alpha \in A}{T}_{\alpha }$, with the product topology

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3: Structured Description 2

Here is the rules of Structured Description.

Entities:
$\{T_1,...,T_n\}$: $T_j\in \{\text{ the topological spaces }\}$
$\ast T$: $=T_1\times ...\times T_n$ with the product topology
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Conditions:
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4: Natural Language Description 2

For any topological spaces, $\{T_1,...,T_n\}$, the product set, $T:=T_1\times ...\times T_n$, with the product topology

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References

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