610: Top2 Category

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definition of $To{p}^2$ category

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Topics

About: category

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of category.
• The reader knows a definition of topological subspace.
• The reader knows a definition of continuous map.

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Target Context

• The reader will have a definition of $To{p}^2$ category.

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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

Here is the rules of Structured Description.

Entities:
$\ast To{p}^2$: $\in \{\text{ the categories }\}$
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Conditions:
$Obj(To{p}^2)=\{\text{ the pairs of topological space and any subspace of the space }\}$
$\wedge$
$\mathrm{\forall }{O}_1=(T_1^{\prime },T_1),O_2=(T_2^{\prime },T_2)\in Obj(To{p}^2)(Mor(O_1,O_2)=\{f:T_1^{\prime }\to T_2^{\prime }|f\in \{\text{ the continuous maps such that }f(T_1)\subseteq T_2\}\})$
$\wedge$
$\mathrm{\forall }{O}_1=(T_1^{\prime },T_1),O_2=(T_2^{\prime },T_2),O_3=(T_3^{\prime },T_3)\in Obj(To{p}^2),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$.
//

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2: Natural Language Description

The category, $To{p}^2$, such that $Obj(To{p}^2)=\{\text{ the pairs of topological space and any subspace of the space }\}$, $\mathrm{\forall }{O}_1=(T_1^{\prime },T_1),O_2=(T_2^{\prime },T_2)\in Obj(To{p}^2)(Mor(O_1,O_2)=\{f:T_1^{\prime }\to T_2^{\prime }|f\in \{\text{ the continuous maps such that }f(T_1)\subseteq T_2\}\})$, and $\mathrm{\forall }{O}_1=(T_1^{\prime },T_1),O_2=(T_2^{\prime },T_2),O_3=(T_3^{\prime },T_3)\in Obj(To{p}^2),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$

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3: Note

"$f_2\circ f_1=f_2\circ f_1$" may seem trivial, but the left hand side denotes the composition of the morphisms and the right hand side denotes the composition of the maps, which is not trivial.

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References

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