In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.

Definitions

Neighbourhood of a point

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 ${\displaystyle p}$,

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.

A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points. A closed rectangle, as illustrated in the figure, is not a neighbourhood of all its points; points on the edges or corners of the rectangle are not contained in any open set that is contained within the rectangle.

The collection of all neighbourhoods of a point is called the neighbourhood system at the point.

Neighbourhood of a set

If ${\displaystyle S}$ is a subset of a topological space ${\displaystyle X}$, then a neighbourhood of ${\displaystyle S}$ is a set ${\displaystyle V}$ that includes an open set ${\displaystyle U}$ containing ${\displaystyle S}$,

It follows that a set ${\displaystyle V}$ is a neighbourhood of ${\displaystyle S}$ if and only if it is a neighbourhood of all the points in ${\displaystyle S.}$ Furthermore, ${\displaystyle V}$ is a neighbourhood of ${\displaystyle S}$ if and only if ${\displaystyle S}$ is a subset of the interior of ${\displaystyle V.}$ A neighbourhood of ${\displaystyle S}$ that is also an open subset of ${\displaystyle X}$ is called an open neighbourhood of ${\displaystyle S.}$ The neighbourhood of a point is just a special case of this definition.

In a metric space

In a metric space ${\displaystyle M=(X,d),}$ a set ${\displaystyle V}$ is a neighbourhood of a point ${\displaystyle p}$ if there exists an open ball with center ${\displaystyle p}$ and radius ${\displaystyle r>0,}$ such that

is contained in ${\displaystyle V.}$

${\displaystyle V}$ is called a uniform neighbourhood of a set ${\displaystyle S}$ if there exists a positive number ${\displaystyle r}$ such that for all elements ${\displaystyle p}$ of ${\displaystyle S,}$

is contained in ${\displaystyle V.}$

Under the same condition, for ${\displaystyle r>0,}$ the ${\displaystyle r}$-neighbourhood ${\displaystyle S_{r}}$ of a set ${\displaystyle S}$ is the set of all points in ${\displaystyle X}$ that are at distance less than ${\displaystyle r}$ from ${\displaystyle S}$ (or equivalently, ${\displaystyle S_{r}}$ is the union of all the open balls of radius ${\displaystyle r}$ that are centered at a point in ${\displaystyle S}$):