definition of locally finite set of subsets of topological space
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of power set of set.
- The reader knows a definition of neighborhood of point.
Target Context
- The reader will have a definition of locally finite set of subsets of topological space.
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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\(*S\): \(\in Pow (T)\)
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Conditions:
\(\forall p \in T (\exists N_p \subseteq T \in \{\text{ the neighborhoods of } p\} (\{S' \in S \vert S'\cap N_p \neq \emptyset\} \in \{\text{ the finite sets }\}))\)
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2: Natural Language Description
For any topological space, \(T\), any set of subsets, \(S \in Pow (T)\), such that for each \(p \in T\), there is a neighborhood, \(N_p \subseteq T\), of \(p\), such that only finite number of elements of \(S\) intersects \(N_p\)
