Neighbourhood (mathematics)

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.

A set {\displaystyle V}V in the plane is a neighbourhood of a point {\displaystyle p}p if a small disc around {\displaystyle p}p is contained in {\displaystyle V.}V.

1 Definitions

1.1 Neighbourhood of a point

If {\displaystyle X}X is a topological space and {\displaystyle p}p is a point in {\displaystyle X,}X, then a neighbourhood of {\displaystyle p}p is a subset {\displaystyle V}V of {\displaystyle X}X that includes an open set {\displaystyle U}U containing {\displaystyle p}p,

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

The neighbourhood {\displaystyle V}V need not be an open subset {\displaystyle X,}X, but when {\displaystyle V}V is open in {\displaystyle X}X then it is called an open neighbourhood.[1] Some authors have been known to require neighbourhoods to be open, so it is important to note 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 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.

A closed rectangle does not have a neighbourhood on any of its corners or its boundary.

1.2 Neighbourhood of a set

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

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

2 In a metric space

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

{\displaystyle B_{r}§=B(p;r)={x\in X:d(x,p) is contained in {\displaystyle V.}V.
{\displaystyle V}V is called uniform neighbourhood of a set {\displaystyle S}S if there exists a positive number {\displaystyle r}r such that for all elements {\displaystyle p}p of {\displaystyle S,}S,

{\displaystyle B_{r}§={x\in X:d(x,p) is contained in {\displaystyle V.}V.
For {\displaystyle r>0,}{\displaystyle r>0,} the {\displaystyle r}r-neighbourhood {\displaystyle S_{r}}S_r of a set {\displaystyle S}S is the set of all points in {\displaystyle X}X that are at distance less than {\displaystyle r}r from {\displaystyle S}S (or equivalently, {\displaystyle S_{r}}S_r is the union of all the open balls of radius {\displaystyle r}r that are centered at a point in {\displaystyle S}S):

{\displaystyle S_{r}=\bigcup \limits {p\in {}S}B{r}§.}{\displaystyle S_{r}=\bigcup \limits {p\in {}S}B{r}§.}
It directly follows that an {\displaystyle r}r-neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an {\displaystyle r}r-neighbourhood for some value of {\displaystyle r.}r.

A set {\displaystyle S}S in the plane and a uniform neighbourhood {\displaystyle V}V of {\displaystyle S.}S.

The epsilon neighbourhood of a number {\displaystyle a}a on the real number line.

3 Examples

Given the set of real numbers {\displaystyle \mathbb {R} }\mathbb {R} with the usual Euclidean metric and a subset {\displaystyle V}V defined as

{\displaystyle V:=\bigcup _{n\in \mathbb {N} }B\left(n,;,1/n\right),}{\displaystyle V:=\bigcup _{n\in \mathbb {N} }B\left(n,;,1/n\right),}
then {\displaystyle V}V is a neighbourhood for the set {\displaystyle \mathbb {N} }\mathbb {N} of natural numbers, but is not a uniform neighbourhood of this set.

The set M is a neighbourhood of the number a, because there is an ε-neighbourhood of a which is a subset of M.

4 Topology from neighbourhoods

See also: Filters in topology, Topological space § Neighborhood definition, and Axiomatic foundations of topological spaces § Definition via neighbourhoods
The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.

A neighbourhood system on {\displaystyle X}X is the assignment of a filter {\displaystyle N(x)}N(x) of subsets of {\displaystyle X}X to each {\displaystyle x}x in {\displaystyle X,}X, such that

the point {\displaystyle x}x is an element of each {\displaystyle U}U in {\displaystyle N(x)}N(x)
each {\displaystyle U}U in {\displaystyle N(x)}N(x) contains some {\displaystyle V}V in {\displaystyle N(x)}N(x) such that for each {\displaystyle y}y in {\displaystyle V,}V, {\displaystyle U}U is in {\displaystyle N(y).}{\displaystyle N(y).}
One can show that both definitions are compatible, that is, the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.

5 Uniform neighbourhoods

In a uniform space {\displaystyle S=(X,\Phi ),}{\displaystyle S=(X,\Phi ),} {\displaystyle V}V is called a uniform neighbourhood of {\displaystyle P}P if there exists an entourage {\displaystyle U\in \Phi }{\displaystyle U\in \Phi } such that {\displaystyle V}V contains all points of {\displaystyle X}X that are {\displaystyle U}U-close to some point of {\displaystyle P;}{\displaystyle P;} that is, {\displaystyle U[x]\subseteq V}{\displaystyle U[x]\subseteq V} for all {\displaystyle x\in P.}{\displaystyle x\in P.}

6 Deleted neighbourhood

A deleted neighbourhood of a point {\displaystyle p}p (sometimes called a punctured neighbourhood) is a neighbourhood of {\displaystyle p,}p, without {\displaystyle {p}.}{\displaystyle {p}.} For instance, the interval {\displaystyle (-1,1)={y:-1

7 See also

Neighbourhood system
Region (mathematics)
Tubular neighbourhood