611: Top∗ Category

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definition of $To{p}^{\ast }$ 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 space.
• The reader knows a definition of continuous map.

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Target Context

• The reader will have a definition of $To{p}^{\ast }$ 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:
$\ast To{p}^{\ast }$: $\in \{\text{ the categories }\}$
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Conditions:
$Obj(To{p}^{\ast })=\{\text{ the pairs of topological space and any point on the space }\}$.
$\wedge$
$\mathrm{\forall }{O}_1=(T_1,p_1),O_2=(T_2,p_2)\in Obj(To{p}^{\ast })(Mor(O_1,O_2)=\{f:T_1\to T_2|f\in \text{ the continuous maps such that }f(p_1)=p_2\})$.
$\wedge$
$\mathrm{\forall }{O}_1=(T_1,p_1),O_2=(T_2,p_2),O_3=(T_3,p_3)\in Obj(To{p}^{\ast }),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$.
//

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2: Natural Language Description

The category, $To{p}^{\ast }$, such that $Obj(To{p}^{\ast })=\{\text{ the pairs of topological space and any point on the space }\}$, $\mathrm{\forall }{O}_1=(T_1,p_1),O_2=(T_2,p_2)\in Obj(To{p}^{\ast })(Mor(O_1,O_2)=\{f:T_1\to T_2|f\in \text{ the continuous maps such that }f(p_1)=p_2\})$, and $\mathrm{\forall }{O}_1=(T_1,p_1),O_2=(T_2,p_2),O_3=(T_3,p_3)\in Obj(To{p}^{\ast }),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$

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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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