definition of \(Top^*\) category
Topics
About: category
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
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.
Target Context
- The reader will have a definition of \(Top^*\) category.
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
Here is the rules of Structured Description.
Entities:
\(*Top^*\): \(\in \{\text{ the categories }\}\)
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Conditions:
\(Obj (Top^*) = \{\text{ the pairs of topological space and any point on the space }\}\).
\(\land\)
\(\forall O_1 = (T_1, p_1), O_2 = (T_2, p_2) \in Obj (Top^*) (Mor (O_1, O_2) = \{f: T_1 \to T_2 \vert f \in \text{ the continuous maps such that } f (p_1) = p_2\})\).
\(\land\)
\(\forall O_1 = (T_1, p_1), O_2 = (T_2, p_2), O_3 = (T_3, p_3) \in Obj (Top^*), \forall f_1 \in Mor (O_1, O_2), \forall f_2 \in Mor (O_2, O_3) (f_2 \circ f_1 = f_2 \circ f_1)\).
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2: Natural Language Description
The category, \(Top^*\), such that \(Obj (Top^*) = \{\text{ the pairs of topological space and any point on the space }\}\), \(\forall O_1 = (T_1, p_1), O_2 = (T_2, p_2) \in Obj (Top^*) (Mor (O_1, O_2) = \{f: T_1 \to T_2 \vert f \in \text{ the continuous maps such that } f (p_1) = p_2\})\), and \(\forall O_1 = (T_1, p_1), O_2 = (T_2, p_2), O_3 = (T_3, p_3) \in Obj (Top^*), \forall f_1 \in Mor (O_1, O_2), \forall f_2 \in Mor (O_2, O_3) (f_2 \circ f_1 = f_2 \circ f_1)\)
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.
