definition of neighborhood of point on topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of open subset of topological space.
Target Context
- The reader will have a definition of neighborhood of point on 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:
\( T\): \(\in \{\text{ the topological spaces }\}\), with any topology, \(O\)
\( t\): \(\in T\)
\(*N_t\): \(\subseteq T\)
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Conditions:
\(\exists U_t \in O (t \in U_t \subseteq N_t)\)
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2: Note
\(N_t\) does not need to be any open subset, although it needs to contain an open subset.
When \(N_t\) is open, it is called "open neighborhood of \(t\) on \(T\)": \(U_t\) is an open neighborhood of \(t\) on \(T\).
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. But in some cases, whether it is a (not-necessarily-open) neighborhood or an open neighborhood matters: for example, when there is a compact neighborhood of a point, there may not be any compact open neighborhood of the point.
