Neighbourhood (mathematics)

Neighbourhood of a point[edit]

If ${\displaystyle X}$ is a topological space and ${\displaystyle p}$ is a point in ${\displaystyle X,}$ then a neighbourhood^[1] of ${\displaystyle p}$ is a subset ${\displaystyle V}$ of ${\displaystyle X}$ that includes an open set ${\displaystyle U}$ containing ${\displaystyle p}$, $${\displaystyle p\in U\subseteq V\subseteq X.}$$


This is equivalent to the point ${\displaystyle p\in X}$ belonging to the topological interior of ${\displaystyle V}$ in ${\displaystyle X.}$


The neighbourhood ${\displaystyle V}$ need not be an open subset of ${\displaystyle X.}$ When ${\displaystyle V}$ is open (resp. closed, compact, etc.) in ${\displaystyle X,}$ it is called an open neighbourhood^[2] (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors^[3] require neighbourhoods to be open, so it is important to note their conventions.