A definition of neighborhood of point
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 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: Definition
For any topological space, \(T\), and its any point, \(p \in T\), any subset, \(N \subseteq T\), that contains an open set, \(U \subseteq T\), that contains \(p\), which is \(p \in U \subseteq N \subseteq T\)
2: 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.
