definition of neighborhood of point
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Definition
- 3: Note
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of open set.
Target Context
- The reader will have a definition of neighborhood of point.
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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( p\): \(\in T\)
\(*N_p\): \(: \subseteq T\)
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Conditions:
\(\exists U_p \in \{\text{ the open subsets of } T\} (p \in U_p \subseteq N_p)\)
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2: Natural Language Definition
For any topological space, \(T\), and its any point, \(p \in T\), any subset, \(N_p \subseteq T\), that contains an open set, \(U_p \subseteq T\), that contains \(p\), which is \(p \in U_p \subseteq N_p \subseteq T\)
3: Note
Some people mean 'open neighborhood' by 'neighborhood', while in many cases, whether it is a (not-necessarily-open) neighborhood or an open neighborhood does not matter, because, for example, "there is a neighborhood" equals "there is an open neighborhood", because if there is a neighborhood, there is an open neighborhood in it, and if there is an open neighborhood, there is the neighborhood that is the open neighborhood.
